QuestionIn the following system of linear equations, k is a constant and x and y are variables. For what value of k will the system of equations have no solution?
kx - 3y = 4
4x - 5y = 7Hint:1. An equation where no value can be substituted for the variable which will satisfy the equation i.e. for no value of x (or any other variable in the equation), would LHS equal RHS.
Hence, an
equation where LHS ≠ RHS for any value of x has no solutions.
2. For Equations with No solutions:-
(a1/a2) = (b1/b2) ≠ (c1/c2) The correct answer is: For k = 2.4, the given set of equations will have no solution
Step-by-step solution:-
From the given information, we have-
kx - 3y = 4 ….................................................. (Equation i)
4x - 5y = 7 ….................................................. (Equation ii)
Comparing the above equations with standard form of equations i.e. ax + by = c, we get-
a1 = k, b1 = -3, c1 = 4
a2 = 4, b2 = -5, c2 = 7
We know that for Equations with no solution-
(a1/a2) = (b1/b2) ≠
(c1/c2)
i.e. k/4 = -3/-5 ≠ 4/7 .................................................. (Equation iii)
∴ k/4 = -3/-5 ........................................................................... (From Equation iii)
∴ k/4 = 3/5
∴ 5k = 4 * 3 .......................................................................... (Cross multiplying)
∴ 5k = 12
∴ k = 12/5
∴ k = 2.4
Final Answer:-
∴ For k = 2.4, the given set of equations will have no solution.Book A Free DemoMobile* +91 I agree to get WhatsApp notifications & Marketing updates Maths- Determine whether the following equations have unique solution y = 2x + 5
y = 3x – 2.Hint:-
An Equation is said to have Unique solution when there exists only one point, on substituting which, LHS = RHS.
For Equations with Unique solution (One solution):-
(a1/a2) ≠ (b1/b2)
Where a1, a2 are
the coefficients of variable x in both the equations i & ii, respectively
and b1, b2 are the coefficients of variable y in both the equations i & ii, respectively.
Step-by-step solution:-
y = 2x + 5
∴ -2x + y = 5 ….............................................. (Taking variables and constants on either sides of the equation)
i.e. 2x - y = -5 …......................... (Multiplying both sides by -1)
....................................................................................... (Equation i)
y = 3x – 2
∴ -3x + y = -2 ….............................................. (Taking variables and constants on either sides of the equation)
i.e. 3x - y = 2 …......................... (Multiplying both sides by -1) ....................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear
equation in 2 variables, i.e. ax + by = c, we get-
a1 = 2; b1 = -1; c1 = -5 and
a2 = 3; b2 = -1; c2 = 2
Now, we calculate a1/a2 and b1/b2 and check whether the 2 are equal or not.
a1/a2 = 2/3 = 0.67 ............................................................................................................................................................ (Equation iii)
b1/b2 = -1/-1 = 1
.............................................................................................................................................................. (Equation iv)
From Equations iii & iv, We observe that-
a1/a2 ≠ b1/b2 for the given Equations.
∴ The given Equations have Unique solution.
Final Answer:-
∴ The given equations y = 2x + 5 and y = 3x – 2 have unique solution. Determine whether the following equations have unique solution y = 2x + 5
y
= 3x – 2.Maths-General Hint:-
An Equation is said to have Unique solution when there exists only one point, on substituting which, LHS = RHS.
For Equations with Unique solution (One solution):-
(a1/a2) ≠ (b1/b2)
Where a1, a2 are the coefficients of variable x in both the equations i & ii, respectively
and b1, b2 are the coefficients of variable y in both the equations i & ii, respectively.
Step-by-step solution:-
y = 2x +
5
∴ -2x + y = 5 ….............................................. (Taking variables and constants on either sides of the equation)
i.e. 2x - y = -5 …......................... (Multiplying both sides by -1) ....................................................................................... (Equation i)
y = 3x – 2
∴ -3x + y = -2 ….............................................. (Taking variables and constants on either sides of the equation)
i.e. 3x - y = 2
…......................... (Multiplying both sides by -1) ....................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 2; b1 = -1; c1 = -5 and
a2 = 3; b2 = -1; c2 = 2
Now, we calculate a1/a2 and b1/b2 and check whether the 2 are equal or not.
a1/a2 = 2/3 = 0.67
............................................................................................................................................................ (Equation iii)
b1/b2 = -1/-1 = 1 .............................................................................................................................................................. (Equation iv)
From Equations iii & iv, We observe that-
a1/a2 ≠ b1/b2 for the given Equations.
∴ The given Equations have Unique
solution.
Final Answer:-
∴ The given equations y = 2x + 5 and y = 3x – 2 have unique solution. Maths- Solve the following equation : 2(1 - x) + 5x = 3(x + 1) and say whether it is having one solution , no solution or infinitely many solutions ?
Hint:-
An equation where no value can be substituted for the variable which will satisfy the equation i.e. for no value of x (or any other variable in the equation), would LHS equal RHS.
Hence, an equation where LHS ≠ RHS for any value of x has no solutions.
We will simplify the given equation and check whether LHS = RHS or not.
Step-by-step solution:-
Simplifying the given equation i.e. 2(1 - x) + 5x = 3(x + 1), we get-
2(1 - x) + 5x = 3(x + 1)
∴ 2 - 2x + 5x = 3x +
3
∴ -2x + 5x - 3x = 3 - 2
∴ 0 ≠ 1
∴ LHS ≠ RHS
Since in the above equation, LHS is not equal to RHS, the given equation has no solutions.
Final Answer:-
∴ The given equation 2(1 - x) + 5x = 3(x + 1) can be classified as having no solution. Solve the following equation : 2(1 - x) + 5x = 3(x + 1) and say whether it is having one solution , no solution or infinitely many solutions ?Maths-General Hint:-
An equation
where no value can be substituted for the variable which will satisfy the equation i.e. for no value of x (or any other variable in the equation), would LHS equal RHS.
Hence, an equation where LHS ≠ RHS for any value of x has no solutions.
We will simplify the given equation and check whether LHS = RHS or not.
Step-by-step solution:-
Simplifying the given equation i.e. 2(1 - x) + 5x = 3(x + 1), we get-
2(1 - x) + 5x = 3(x + 1)
∴ 2 - 2x + 5x = 3x + 3
∴ -2x + 5x - 3x = 3 -
2
∴ 0 ≠ 1
∴ LHS ≠ RHS
Since in the above equation, LHS is not equal to RHS, the given equation has no solutions.
Final Answer:-
∴ The given equation 2(1 - x) + 5x = 3(x + 1) can be classified as having no solution. Maths- Given five noncollinear
points, make a conjecture about the number of ways to connect different pairs of points.Hint:
Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture.
Solution
Let’s first make a table and then look for the pattern The sequence of the number of connections is 0, 1, 3, 6, 10, ….
We can see that
1 = 0 + 1
3 = 1 + 2
6 = 3 + 3
So by
seeing the pattern the number of connections in 6 points is given as
6 + 4 = 10
So by seeing the pattern the conjecture that can be made is that the number of ways to connect the 5 collinear points in different pairs of points is 10
Final Answer:
Hence, the conjecture that can be concluded is “the number of ways to connect the 6 collinear points in different pairs of points is 10”. Given five noncollinear points, make a conjecture about the
number of ways to connect different pairs of points.Maths-General Hint:
Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture.
Solution
Let’s first make a table and then look for the pattern The sequence of the number of connections is 0, 1, 3, 6, 10, ….
We can see that
1 = 0 + 1
3 = 1 + 2
6 =
3 + 3
So by seeing the pattern the number of connections in 6 points is given as
6 + 4 = 10
So by seeing the pattern the conjecture that can be made is that the number of ways to connect the 5 collinear points in different pairs of points is 10
Final Answer:
Hence, the conjecture that can be concluded is “the number of ways to connect the 6 collinear points in different pairs of points is 10”. Maths- Check, whether the following equation has exactly one solution or infinitely many solution or no solution. 4x + 2 = 4x – 5Hint:-
An equation where no value can be substituted for the variable which will satisfy the equation i.e. for no value of x (or any other
variable in the equation), would LHS equal RHS.
Hence, an equation where LHS ≠ RHS for any value of x has no solutions.
We will simplify the given equation and check whether LHS = RHS or not.
Step-by-step solution:-
Simplifying the given equation i.e. 4x + 2 = 4x - 5, we get-
4x + 2 = 4x - 5
∴ 4x - 4x + 2 = 4x - 4x - 5 ........................................................... (Adding -4x both the sides)
∴ 0 + 2 = -5
∴ 2 ≠ -5
∴ LHS ≠ RHS
Since in the above
equation, LHS is not equal to RHS, the given equation has no solutions.
Final Answer:-
∴ The given equation 4x + 2 = 4x - 5 can be classified as having no solution. Check, whether the following equation has exactly one solution or infinitely many solution or no solution. 4x + 2 = 4x – 5Maths-General Hint:-
An equation where no value can be substituted for the variable which will satisfy the equation i.e. for no value of x
(or any other variable in the equation), would LHS equal RHS.
Hence, an equation where LHS ≠ RHS for any value of x has no solutions.
We will simplify the given equation and check whether LHS = RHS or not.
Step-by-step solution:-
Simplifying the given equation i.e. 4x + 2 = 4x - 5, we get-
4x + 2 = 4x - 5
∴ 4x - 4x + 2 = 4x - 4x - 5 ........................................................... (Adding -4x both the sides)
∴ 0 + 2 = -5
∴ 2 ≠ -5
∴ LHS ≠ RHS
Since in the
above equation, LHS is not equal to RHS, the given equation has no solutions.
Final Answer:-
∴ The given equation 4x + 2 = 4x - 5 can be classified as having no solution. Maths- Write the converse and biconditional statement for the given conditional
statement.
If a triangle is equilateral, then it is equiangular.Hint:
The converse of a statement is formed by switching the hypothesis and the conclusion. The converse of "if p, then q" is "if q, then p."
A biconditional statement is a logic statement that includes the phrase, "if and
only if," sometimes abbreviated as "iff." The logical biconditional statement can be written as p iff q, p if and only if q, p ↔ q
Solution
It is given
p: Triangle is equilateral.
q: Triangle is equiangular.
Conditional statement of p → q is given as“If a triangle is equilateral, then it is equiangular. ”
Converse statement of p → q is q → p and it is written as “ If a triangle is equiangular, then it is equilateral. ”
Biconditional statement of p → q is p ↔ q“
Triangle is equiangular, if and only if Triangle is equiangular. ”
Final Answer:
Converse statement: If a triangle is equiangular, then it is equilateral.
Biconditional statement: Triangle is equiangular, if and only if Triangle is equiangular. . Write the converse and biconditional statement for the given conditional statement.
If a triangle is equilateral, then it is equiangular.Maths-General
Hint:
The converse of a statement is formed by switching the hypothesis and the conclusion. The converse of "if p, then q" is "if q, then p."
A biconditional statement is a logic statement that includes the phrase, "if and only if," sometimes abbreviated as "iff." The logical biconditional
statement can be written as p iff q, p if and only if q, p ↔ q
Solution
It is given
p: Triangle is equilateral.
q: Triangle is equiangular.
Conditional statement of p → q is given as“If a triangle is equilateral, then it is equiangular. ”
Converse statement of p → q is q → p and it is written as “ If a triangle is equiangular, then it is equilateral. ”
Biconditional statement of p → q is p ↔ q“ Triangle is equiangular, if and only if Triangle is equiangular.
”
Final Answer:
Converse statement: If a triangle is equiangular, then it is equilateral.
Biconditional statement: Triangle is equiangular, if and only if Triangle is equiangular. . Maths- Check, whether the following
equation has exactly one solution or infinitely many solution or no solution. 4x - 5 = 2(2x - 1) – 3Hint:-
When Both sides of an equation are equal, the said equation is said to have infinitely many solutions.
i.e. When LHS = RHS, the given equation has infinitely many solutions.
Step-by-step solution:-
Simplifying the given equation i.e. 4x - 5 = 2(2x - 1) – 3, we get-
4x - 5 = 2(2x - 1) - 3
∴ 4x - 5 = 4x - 2 - 3 ...................................... (Opening the
bracket and multiplying 2 with the entire expression on RHS)
∴ 4x - 4x = -2 - 3 + 5 ....................................... (Taking variables and constants on either sides of the equation)
∴ 0 = 0
∴ LHS = RHS
Since in the above equation, LHS is equal to RHS, the given equation has Infinitely Many Solutions.
Final Answer:-
∴ The given equation 4x - 5 = 2(2x - 1) - 3 has Infinitely Many Solutions. Check, whether the following equation has exactly one solution or
infinitely many solution or no solution. 4x - 5 = 2(2x - 1) – 3Maths-General Hint:-
When Both sides of an equation are equal, the said equation is said to have infinitely many solutions.
i.e. When LHS = RHS, the given equation has infinitely many solutions.
Step-by-step solution:-
Simplifying the given equation i.e. 4x - 5 = 2(2x - 1) – 3, we get-
4x - 5 = 2(2x - 1) - 3
∴ 4x - 5 = 4x - 2 - 3 ......................................
(Opening the bracket and multiplying 2 with the entire expression on RHS)
∴ 4x - 4x = -2 - 3 + 5 ....................................... (Taking variables and constants on either sides of the equation)
∴ 0 = 0
∴ LHS = RHS
Since in the above equation, LHS is equal to RHS, the given equation has Infinitely Many Solutions.
Final Answer:-
∴ The given equation 4x - 5 = 2(2x - 1) - 3 has Infinitely Many Solutions. Maths- The system of equations 3x - 5y = 20 ; 6x - 10y = 40 hasHint:-
1. For Equations with Unique solution (One solution):-
(a1/a2) ≠ (b1/b2)
2. For Equations with Infinitely Many solutions:-
(a1/a2) = (b1/b2) = (c1/c2)
3. For Equations with No
solutions:-
(a1/a2) = (b1/b2) ≠ (c1/c2)
Step-by-step solution:-
3x - 5y = 20 ....................................................................................... (Equation i)
6x - 10y = 40 ..................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 3; b1 = -5; c1 = 20 and
a2 = 6; b2 = -10; c2 = 40
Now, we
calculate a1/a2; b1/b2 and c1/c2 to check the corelation between the 3:-
a1/a2 = 3/6 = 1/2 ............................................................................................................................................................ (Equation iii)
b1/b2 = -5/-10 = 1/2 ....................................................................................................................................................... (Equation iv)
c1/c2 = 20/40 = 1/2
......................................................................................................................................................... (Equation v)
From Equations iii, iv & v, We observe that-
a1/a2 = b1/b2 = c1/c2 for the given Equations.
∴ The given Equations have Infinitely Many Solutions.
Final Answer:-
∴ Option c i.e. Infinitely Many Solutions is the correct answer. The system of equations 3x - 5y = 20 ; 6x - 10y = 40
hasMaths-General Hint:-
1. For Equations with Unique solution (One solution):-
(a1/a2) ≠ (b1/b2)
2. For Equations with Infinitely Many solutions:-
(a1/a2) = (b1/b2) = (c1/c2)
3. For Equations with No solutions:-
(a1/a2) = (b1/b2) ≠ (c1/c2)
Step-by-step solution:-
3x - 5y = 20 ....................................................................................... (Equation i)
6x - 10y = 40
..................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 3; b1 = -5; c1 = 20 and
a2 = 6; b2 = -10; c2 = 40
Now, we calculate a1/a2; b1/b2 and c1/c2 to check the corelation between the 3:-
a1/a2 = 3/6 = 1/2
............................................................................................................................................................ (Equation iii)
b1/b2 = -5/-10 = 1/2 ....................................................................................................................................................... (Equation iv)
c1/c2 = 20/40 = 1/2
......................................................................................................................................................... (Equation v)
From Equations iii, iv & v, We observe that-
a1/a2 = b1/b2 = c1/c2 for the given Equations.
∴ The given Equations have Infinitely Many Solutions.
Final Answer:-
∴ Option c i.e. Infinitely Many Solutions is the correct answer. Maths- Describe the pattern in the numbers. Write the next number in the pattern.
7, 3.5, 1.75, 0.875, …Hint:
A pattern is a repeated arrangement of numbers, shapes, colours etc. The Pattern can be related to any type of event or object. If the set of numbers are related to
each other in a specific rule, then the rule or manner is called a pattern.
Solution
The sequence is given: 7, 3.5, 1.75, 0.875, …
We can see that So we can conclude that the nth number will be the result of the division of (n-1)th number with 2 or we can say that every number is the division of the previous number with 2.
The next number will be
= 0.4375
Final Answer:
Hence, the next number of the sequence will be
0.4375 and the pattern here is that every number is the division of the previous number with 2. Describe the pattern in the numbers. Write the next number in the pattern.
7, 3.5, 1.75, 0.875, …Maths-General Hint:
A pattern is a repeated arrangement of numbers, shapes, colours etc. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific
rule, then the rule or manner is called a pattern.
Solution
The sequence is given: 7, 3.5, 1.75, 0.875, …
We can see that So we can conclude that the nth number will be the result of the division of (n-1)th number with 2 or we can say that every number is the division of the previous number with 2.
The next number will be
= 0.4375
Final Answer:
Hence, the next number of the sequence will be 0.4375 and the pattern
here is that every number is the division of the previous number with 2. Maths- Check, whether the following equation has exactly one solution or infinitely many solution or no solution. 4x - 3 = 2x + 13Hint:-
When a given equation has only
1 value that can be substituted for the variable which will satisfy the given equation, the said equation has only 1 solution. i.e. after simplifying the equation, we will get a single value as the solution for x.
We will simplify the given equation and the number of values we get for the variable.
Step-by-step solution:-
Simplifying the given equation i.e. 4x - 3 = 2x + 13, we get-
4x - 3 = 2x + 13
∴ 4x - 2x = 13 + 3
...........................................................................(Taking variables and constants on either side of the equation)
∴ 2x = 16
∴ x = 8 …......................................................................... (Dividing both sides by 2)
Since after simplifying the given equation, we get a single value of x i.e. 8 that will satisfy the given equation, we can conclude that the given equation has 1 solution.
Final Answer:-
∴ The given equation i.e. 4x - 3 = 2x +
13 can be classified as an equation with only one solution. Check, whether the following equation has exactly one solution or infinitely many solution or no solution. 4x - 3 = 2x + 13Maths-General Hint:-
When a given equation has only 1 value that can be substituted for the variable which will satisfy the given equation, the said equation has only 1 solution. i.e. after simplifying the equation, we will get a single value as the
solution for x.
We will simplify the given equation and the number of values we get for the variable.
Step-by-step solution:-
Simplifying the given equation i.e. 4x - 3 = 2x + 13, we get-
4x - 3 = 2x + 13
∴ 4x - 2x = 13 + 3 ...........................................................................(Taking variables and constants on either side of the equation)
∴ 2x = 16
∴ x = 8 …......................................................................... (Dividing both sides
by 2)
Since after simplifying the given equation, we get a single value of x i.e. 8 that will satisfy the given equation, we can conclude that the given equation has 1 solution.
Final Answer:-
∴ The given equation i.e. 4x - 3 = 2x + 13 can be classified as an equation with only one solution. Maths-
Describe the pattern in the numbers. Write the next number in the pattern.
5, − 2, − 9, − 16, …Hint:
A pattern is a repeated arrangement of numbers, shapes, colours etc. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern.
Solution
The sequence is given: 5, − 2, − 9, − 16, …
We can see that
-2 = 5 - 7
-9 = -2 -
7
-16 = -9 - 7
So we can conclude that the nth number will be the result of the subtraction of (n-1)th number with 7 or we can say that every number is the subtraction of the previous number with 7.
The next number will be
-16 - 7 = -23
Final Answer:
Hence, the next number of the sequence will be -23 and the pattern here is that every number is the subtraction of the previous number with 7. Describe the pattern in the
numbers. Write the next number in the pattern.
5, − 2, − 9, − 16, …Maths-General Hint:
A pattern is a repeated arrangement of numbers, shapes, colours etc. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern.
Solution
The sequence is given: 5, − 2, − 9, − 16, …
We can see that
-2 = 5 -
7
-9 = -2 - 7
-16 = -9 - 7
So we can conclude that the nth number will be the result of the subtraction of (n-1)th number with 7 or we can say that every number is the subtraction of the previous number with 7.
The next number will be
-16 - 7 = -23
Final Answer:
Hence, the next number of the sequence will be -23 and the pattern here is that every number is the subtraction of the previous number with 7. Maths-
Use the Law of Detachment to make a valid conclusion in the true situation. Alex goes to the park every Sunday evening. Today is Sunday.
Hint:
Law of Detachment states that if p q is true and it is given that p is true then we can conclude
that q is also true. Here, the statement is termed as Hypothesis and the statement q is termed as conclusion.
Solution
It is given that Alex goes to the park every Sunday evening or we can say if the day is Sunday then Alex will go to the park in the evening.
p: The day is Sunday
q: Alex will go to the park in the evening
So we can write the given statement “if the day is Sunday then Alex will go to the park in the evening” as We are given that today is Sunday
which means the p statement is true and hence we can conclude that the q statement is also true i.e. Alex will go to the park in the evening.
Final Answer:
Hence, we can conclude that Alex will go to the park in the evening.
Use the Law of Detachment to make a valid conclusion in the true situation. Alex goes to the park every Sunday evening. Today is Sunday.
Maths-General Hint:
Law
of Detachment states that if p q is true and it is given that p is true then we can conclude that q is also true. Here, the statement is termed as Hypothesis and the statement q is termed as conclusion.
Solution
It is given that Alex goes to the park every Sunday evening or we can say if the day is Sunday then Alex will go to the park in the evening.
p: The day is Sunday
q: Alex will go to the park in the evening
So we can write the given statement “if the day
is Sunday then Alex will go to the park in the evening” as We are given that today is Sunday which means the p statement is true and hence we can conclude that the q statement is also true i.e. Alex will go to the park in the evening.
Final Answer:
Hence, we can conclude that Alex will go to the park in the evening. Maths- For which of the values of a and b will the following pair of linear equations has infinitely many solutions x + 2y = 1; (a - b)x + (a + b)y = a + b - 2
a) a = - 3 , b = 1
b) a = 3 , b = 1
c) a = 2 , b = 2
d) None of the aboveHint:-
1. When Both sides of
an equation are equal, the said equation is said to have infinitely many solutions.
i.e. When LHS = RHS, the given equation has infinitely many solutions.
2. For Equations with Infinitely Many solutions:-
(a1/a2) = (b1/b2) = (c1/c2)
Step-by-step solution:-
x = 2y
∴ x - 2y = 0 .................. (Taking variables and constants on each side of the equation) ............................... (Equation i)
y = 2x
∴ y - 2x = 0 .................. (Taking variables and constants
on each side of the equation)
i.e. 2x - y = 0 ..................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 1; b1 = -2; c1 = 0 and
a2 = 2; b2 = -1; c2 = 0
Now, we calculate a1/a2; b1/b2 and c1/c2 to check the corelation between the 3:-
a1/a2 = 1/2
............................................................................................................................................................ (Equation iii)
b1/b2 = -2/-1 = 2 ....................................................................................................................................................... (Equation iv)
c1/c2 = 0/0 = 0
......................................................................................................................................................... (Equation v)
From Equations iii, iv & v, We observe that-
a1/a2 ≠ b1/b2 ≠ c1/c2 for the given Equations.
∴ The given Equations have Unique Solution.
Final Answer:-
∴ The given equation i.e. 4x - 3 = 2x + 13 can be classified as an equation with only one solution. For which of the values of a and b will the
following pair of linear equations has infinitely many solutions x + 2y = 1; (a - b)x + (a + b)y = a + b - 2
a) a = - 3 , b = 1
b) a = 3 , b = 1
c) a = 2 , b = 2
d) None of the aboveMaths-General Hint:-
1. When Both sides of an equation are equal, the said equation is said to have infinitely many solutions.
i.e. When LHS = RHS, the given equation has infinitely many solutions.
2. For Equations with Infinitely Many
solutions:-
(a1/a2) = (b1/b2) = (c1/c2)
Step-by-step solution:-
x = 2y
∴ x - 2y = 0 .................. (Taking variables and constants on each side of the equation) ............................... (Equation i)
y = 2x
∴ y - 2x = 0 .................. (Taking variables and constants on each side of the equation)
i.e. 2x - y = 0 ..................................................................................... (Equation ii)
Comparing Equations i & ii with the
standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 1; b1 = -2; c1 = 0 and
a2 = 2; b2 = -1; c2 = 0
Now, we calculate a1/a2; b1/b2 and c1/c2 to check the corelation between the 3:-
a1/a2 = 1/2 ............................................................................................................................................................ (Equation iii)
b1/b2 = -2/-1 = 2
....................................................................................................................................................... (Equation iv)
c1/c2 = 0/0 = 0 ......................................................................................................................................................... (Equation v)
From Equations iii, iv & v, We observe that-
a1/a2 ≠ b1/b2 ≠ c1/c2 for the given Equations.
∴ The given Equations have Unique
Solution.
Final Answer:-
∴ The given equation i.e. 4x - 3 = 2x + 13 can be classified as an equation with only one solution. Maths- Describe the pattern in the numbers. Write the next number in the pattern.
3, 1, 1/3, 1/9, …Hint:
A pattern is
a repeated arrangement of numbers, shapes, colours etc. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern.
Solution
The sequence is given: 3, 1, 1/3, 1/9, …
We can see that So we can conclude that the nth number will be the result of the multiplication of (n-1)th number and 1/3 or we can say that every number is the
multiplication of the previous number with 1/3.
The next number will be Final Answer:
Hence, the next number of the sequence will be and the pattern here is that every number is the multiplication of the previous number with 1/3. Describe the pattern in the numbers. Write the next number in the pattern.
3, 1, 1/3, 1/9, …Maths-General Hint:
A pattern is a repeated
arrangement of numbers, shapes, colours etc. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern.
Solution
The sequence is given: 3, 1, 1/3, 1/9, …
We can see that So we can conclude that the nth number will be the result of the multiplication of (n-1)th number and 1/3 or we can say that every number is the multiplication
of the previous number with 1/3.
The next number will be Final Answer:
Hence, the next number of the sequence will be and the pattern here is that every number is the multiplication of the previous number with 1/3. Maths- In An academic contest, the Correct answer earns 12 points and the incorrect answer loses 5 points. In
the final round, School A starts with 165 points and gives the same number of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied. Find
b) How many answers did each school get correct in the final round?- Step by step explanation:
○ Step 1:
○ Solve equation: 165 + 12x - 5x = 65 + 12x 165 + 12x - 5x =
65 + 12x
165 + 7x = 65 + 12x
165 - 65 = 12x - 7x
100 = 5x
= x
20 = x
Hence, the school A gives 20 correct answers. In An academic contest, the Correct answer earns 12 points and the incorrect answer loses 5 points. In the final round, School A starts with 165 points and gives the same number of correct and incorrect answers. School B starts with 65 points
and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied. Find
b) How many answers did each school get correct in the final round?Maths-General - Step by step explanation:
○ Step 1:
○ Solve equation: 165 + 12x - 5x = 65 + 12x 165 + 12x - 5x = 65 + 12x
165 + 7x = 65 + 12x
165 - 65 = 12x - 7x
100
= 5x
= x
20 = x
Hence, the school A gives 20 correct answers. Maths- Classify the linear equations x = 2y and, y = 2x as having one solution, no solution or infinitely
many solutions.Hint:-
1. For Equations with Unique solution (One solution):-
(a1/a2) ≠ (b1/b2)
2. For Equations with Infinitely Many solutions:-
(a1/a2) = (b1/b2) = (c1/c2)
3. For Equations with No solution:-
(a1/a2) = (b1/b2) ≠ (c1/c2)
Step-by-step solution:-
x = 2y
∴ x - 2y = 0 .................. (Taking variables and constants on each side of the equation) ............................... (Equation i)
y = 2x
∴ y - 2x = 0 .................. (Taking
variables and constants on each side of the equation)
i.e. 2x - y = 0 ..................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 1; b1 = -2; c1 = 0 and
a2 = 2; b2 = -1; c2 = 0
Now, we calculate a1/a2; b1/b2 and c1/c2 to check the corelation between the 3:-
a1/a2 = 1/2
............................................................................................................................................................ (Equation iii)
b1/b2 = -2/-1 = 2 ....................................................................................................................................................... (Equation iv)
c1/c2 = 0/0 = 0
......................................................................................................................................................... (Equation v)
From Equations iii, iv & v, We observe that-
a1/a2 ≠ b1/b2 ≠ c1/c2 for the given Equations.
∴ The given Equations have Unique Solution.
Final Answer:-
∴ The linear equations x=2y and y=2x have Unique solution. Classify the linear equations x = 2y and, y = 2x as having one solution, no solution
or infinitely many solutions.Maths-General Hint:-
1. For Equations with Unique solution (One solution):-
(a1/a2) ≠ (b1/b2)
2. For Equations with Infinitely Many solutions:-
(a1/a2) = (b1/b2) = (c1/c2)
3. For Equations with No solution:-
(a1/a2) = (b1/b2) ≠ (c1/c2)
Step-by-step solution:-
x = 2y
∴ x - 2y = 0 .................. (Taking variables and constants on each side of the equation) ...............................
(Equation i)
y = 2x
∴ y - 2x = 0 .................. (Taking variables and constants on each side of the equation)
i.e. 2x - y = 0 ..................................................................................... (Equation ii)
Comparing Equations i & ii with the standard form of a linear equation in 2 variables, i.e. ax + by = c, we get-
a1 = 1; b1 = -2; c1 = 0 and
a2 = 2; b2 = -1; c2 = 0
Now, we calculate a1/a2; b1/b2 and c1/c2 to check the corelation between the
3:-
a1/a2 = 1/2 ............................................................................................................................................................ (Equation iii)
b1/b2 = -2/-1 = 2 ....................................................................................................................................................... (Equation iv)
c1/c2 = 0/0 = 0
......................................................................................................................................................... (Equation v)
From Equations iii, iv & v, We observe that-
a1/a2 ≠ b1/b2 ≠ c1/c2 for the given Equations.
∴ The given Equations have Unique Solution.
Final Answer:-
∴ The linear equations x=2y and y=2x have Unique solution.
What is the value of K KX 3y 3?
For which value(s) of k will the pair of equations kx + 3y = k – 3 ; 12x + ky = k have no solution? Therefore, value of k for which the given pair of linear equations has no solution is k = – 6.
For what values of k will be the following pair of linear equations have infinitely many solutions?
For what value of k, the following pair of linear equations has infinitely many solutions? 20x + 10y - k = 0. Hence, the given system of equations has infinitely many solutions, when k = 10.
For what value of k will the following equations have infinitely many solutions KX Y 2 6x 2y 4?
So the value of k must be equals to 3 for the given equation to have infinitely many solutions.
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