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

FAQs on Linear Equations in Two Variables

What are linear equations in two variables?

Linear equations in two variables are algebraic expressions involving two variables, typically denoted as x and y, with no variable raised to a power greater than 1.

How do you graph linear equations in two variables?

To graph a linear equation in two variables, ax + by = c, you can rearrange it into slope-intercept form, y=mx + b, where m is the slope and b is the y-intercept, then plot points using the slope and intercept.

What methods can be used to solve systems of linear equations in two variables?

Systems of linear equations in two variables can be solved using methods such as substitution, elimination, and graphing.

What is the significance of the slope in a linear equation?

The slope of a linear equation represents the rate of change of the dependent variable (y) with respect to the independent variable (x). It indicates the steepness of the line on the graph.

What does it mean when two linear equations in two variables are parallel?

If two linear equations in two variables have the same slope but different y-intercepts, they are parallel lines, and the system of equations has no solution. This indicates that the lines never intersect on the coordinate plane.