Practice Problem on Linear Equations in Two Variables

In this article, we will learn about one interesting topic which is covered in class 9 and class 10 mathematics. We will look at some formulas and problems of Linear equations in two variables.

Important Formulas on Linear Equations in Two Variables

• Linear equations in two variables are expressed in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero.
• The solution of the equation represents the values of x and x for which the equation holds true.
• if (a₁/a₂ ≠ b₁/b₂) then the equation has exactly one solution. The lines are intersecting lines.
• if (a₁/a₂ = b₁/b₂ = c₁/c₂) then the equation has infinitely many solution. The lines are coincidental lines.
• if (a₁/a₂ = b₁/b₂ ≠ c₁/c₂) then the equation has no solution. The lines are parallel lines.
• The slope of a line represented in the form y = mx + c is m, where m is the coefficient of x.

Practice Problems with Solutions

Q1. What are the coefficients of the equation 4x - 10y = 46?

Solution:

To find the coefficients of the equation 4x - 10y = 46,

we need to find the term which is multiplying the variable

So, coefficient of x = 4 and coefficient of y = -10

Q2. What is the constant of the equation 4x - 10y = 46?

Solution:

To find the constant of the equation 4x - 10y = 46,

we need to find the term which is not multiplied with any variable

So, constant of 4x - 10y - 46 = 0 is -46.

Q3. Is x = 3 and y = 10 a solution of the equation -14x + 12y = 30 ?

Solution:

To check if a pair of values (x, y) is a solution of the equation -14x + 12y = 30,

we need to verify that left hand side of equation should be equal to right hand side of equation

i.e. L.H.S = R.H.S

So,

⇒ -14x + 12y = 30

⇒ -14 × 3 + 12 × 10 = 30

⇒ -42 + 120 ≠ 30

So, LHS is not equal to RHS.

So, x = 3 and y = 10 are not the solution of the equation -14x + 12y = 30

Q4. Is x = 3 and y = -10 a solution of the equation 10x + 3y = 0?

Solution:

To check if a pair of values (x, y) is a solution of the equation 10x + 3y = 0,

we need to verify that left hand side of equation should be equal to right hand side of equation

i.e. L.H.S = R.H.S

So,

⇒ 10x + 3y

⇒ 10 × 3 + 3 × (-10)

⇒ 30 - 30

⇒ 0

So, LHS is equal to RHS.

So, x = 3 and y = -10 are the solution of the equation 10x + 3y = 0

Q5. What’s the slope of the line 30x - 6y =3?

Solution:

To find the slope of the line 30x - 6y = 3, follow these steps

First, put the equation in the slope intercept form (y = mx + b)

6y = 30x - 3

y = 5x - 1/2

Now, check the coefficient of x

Here the coefficient of x is 5

So, the slope of the line 30x - 6y = 3 is 5.

Q6. What’s the slope of the line -20x + 10y = 8?

Solution:

To find the slope of the line -20x + 10y = 8, follow these steps

First, put the equation in the slope intercept form (y = mx + b)

10y = 20x + 8

y = 2x + 4/5

Now, check the coefficient of x

Here the coefficient of x is 2

So, the slope of the line 30x - 6y = 3 is 2.

Q7. Two Notebook and one pen cost Rs. 35 and 3 Notebook and four pen cost Rs. 65. Find the cost of Notebook and pen separately.

Solution:

Let's denote the cost of one notebook as N and the cost of one pen as P.

1. Two notebooks and one pen cost Rs. 35:

2N + 1P = 35

2. Three notebooks and four pens cost Rs. 65:

3N + 4P = 65

Let's solve it using the elimination method:

Multiplying the first equation by 4 and the second equation by 1 to eliminate P:

1. 4 * (2N + 1P) = 4 * 35 which gives 8N + 4P = 140

2. 1 * (3N + 4P) = 1 * 65 which gives 3N + 4P = 65

Now, subtracting the second equation from the first equation:

(8N + 4P) - (3N + 4P) = 140 - 65

8N + 4P - 3N - 4P = 75

5N = 75

Dividing both sides by 5:

N = 75/5 = 15

Now that we have found the cost of one notebook N = 15, we can substitute this value into one of the original equations to find the cost of one pen.

From the first equation:

2N + 1P = 35

2(15) + 1P = 35

30 + P = 35

P = 35 - 30

P = 5

So, the cost of one notebook is Rs. 15 and the cost of one pen is Rs. 5.

Q8. Find the solution for the given pair of linear equations.

2x + 3y = 7

4x - 6y = 10

Solution:

To find the number of solution , we check the ratio

a₂/a₁ = 4/2 = 2

and, b₂/b₁ = -6/3 = -2

a₂/a₁ ≠ b₂/b₁

So, it have one solution.

Now, to find the solution

we have two equations

2x + 3y = 7 .....(i)

4x - 6y = 10 ......(ii)

Multiply equation (i) by 2

4x + 6y = 14 .....(iii)

Now add equation (ii) and (iii),

we get 8x = 24

So, x = 3.

Now the value of x in equation (i)

6 + 3y = 7

y = 1/3.

So, x = 3 and y = 1/3.

Q9. Find the solution for the given pair of linear equations.

3x + 2y = 10

6x + 4y = 20

Solution:

To find the number of solution , we check the ratio

a₂/a₁ = 6/3 = 2

and, b₂/b₁ = 4/2 = 2

and, c₂/c₁ = 20/10 = 2

Thus, a₂/a₁ = b₂/b₁ = c₂/c₁ = 2

So, it have infinitely many solution.

Now, to find the solution

we have two equations

3x + 2y = 10 ....(i)

6x + 4y = 20 ....(ii)

As, we observe that both lines are the same line.

So, any point which fall on the line is the solution

like, x = 2 and y = 2

x = 3 and y = 1/2 and many more.

Q10. Find the solution for the given pair of linear equations.

2x + 3y =7

4x + 6y = 15

Solution:

To find the number of solution , we check the ratio

a₂/a₁ = 4/2 = 2

and, b₂/b₁ = 6/3 = 2

and, c₂/c₁ = 15/7

So, a₂/a₁ = b₂/b₁ ≠ c₂/c₁.

So, it have no solution.

Problems on Linear Equations in Two Variables

P1. What are the coefficients of the equation 2x − 5y = 20?

P2. What is the constant of the equation 3x + 7y = −14?

P3. Is x=4 and y=2 a solution of the equation −5x + 3y = 7?

P4. Is x=−3 and y=5 a solution of the equation 8x − 2y = −34?

P5. What’s the slope of the line 6x − 9y = 12?

P6. What’s the slope of the line −4x + 8y = −16?

P7. Three apples and two oranges cost $8, and five apples and four oranges cost $18. Find the cost of an apple and an orange separately.

P8. Find the solution for the given pair of linear equations:

• 3x − 2y = 5
• 6x + 4y = 14

P9. Find the solution for the given pair of linear equations:

• 4x + 3y = 12
• 8x + 6y = 24

P10. Find the solution for the given pair of linear equations:

• 5x − 2y = 10
• 10x + 4y = 20