# Linear Equations Problems

Linear equations are mathematical expression which contain "equal to" sign. Problems based on linear equations help students to clear doubts and boost their skills. We clear your concepts ensuring to stay with you in the long run. We have solved each problem step by step with detailed explanation using additional, subtraction, division and multiplication properties.
At the end of this section, we have provided some practice problems which help you to excel your skill and make you exam ready. Pick a question you want to solve and improve in. Students will find it extremely easy to understand linear equations problems and how to go about solving them.

## Problems on Linear Equations

we have designed linear equations questions and answers to help students to clear doubts and boost exam preparations. This is very helpful resource for students. This set of questions is influenced by NCERT math book.  Let us discuss some questions based on linear equations.

Example 1: Solve for x. 5x + 4 = 4x - 10
Solution:   5x + 4 = 4x - 10
5x - 4x = - 10 - 4

Example 2: Find the equation of a line in slope intercept form and standard form given that the line passes through the points $(-1,2)$ and $(3,5)$.

Solution:

Here, $(x_{1},y_{1})$ = (-1,2) and $(x_{2},y_{2})$ = (3,5)

Therefore the slope:

$m$ = $\frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}$ = $\frac{(5-2)}{(3-(-1))}$ = $\frac{3}{4}$

Now using the point slope formula we can write the equation of the line as follows:

$y\ -\ y_{1}$ = $m(x - x_{1})$

Substituting the values of m and $(x_{1},y_{1})$ we have:

$y-2$ = $\frac{3}{4}$ $(x-(-1))$

Simplifying that we have:

4(y - 2) = 3(x + 1)

4y - 8 = 3x + 3

-3x + 4y = 8 + 3

3x - 4y = -11

3x - 4y + 11 = 0 -> Standard form

We can also convert into slope intercept form:

y - 2 = $\frac{3}{4}$ $(x+1)$

y - 2 = $\frac{3x}{4}$ + $\frac{3}{4}$

y = $\frac{3}{4}$ $x$ + $\frac{3}{4}$ + 2

y = $\frac{3}{4}$ $x$ + $\frac{11}{4}$
This is a Slope intercept form!

Example 3: Solve the set of equations by elimination

x - y = 8

2x - 2y = 16

Solution:

x - y = 8 … (1)

2x - 2y = 16 … (2)

Multuply equation (1) by 2
2x - 2y = 16 … (3)

Subtract equation (3) from equation (2)

2x - 2y = 16

-2x + 2y = -16
_______________
0 = 0

This is true for all values of x, y therefore infinite solutions.

Example 4: Verify that x = 3 is not a solution of the equation, 3x - 5 = 2 + x
Solution: Substituting x = 3, in 3x - 5 = 2 + x, we get,
3 (3) - 5 = 2 + 3
=> 9 - 5 = 5
=> 4 = 5, which is not true
Therefore, x = 3 is not a solution of the given equation.

Example 5
: The sum of two numbers is 45 and their ratio is 7 : 8. Find the numbers.

Solution:
Step 1: Translate into symbolic language
Let one of the number be x.
Therefore, the other number will be ( 45 - x )
It is given that the ratio of the two numbers = 7 : 8
Hence we have,  x : ( 45 - x ) = 7 : 8

Expressing it into fraction, we have, $\frac{x}{45-x}$ = $\frac{7}{8}$
Step 2: Solving the equation

$\frac{x}{45-x}$= $\frac{7}{8}$

8 x = 7 ( 45 - x ) [ cross multiplying ]
8x = 315 - 7x
8x + 7x = 315
15 x = 315
$\frac{15x}{15}$ = $\frac{315}{15}$
x = 21
Step 3: Interpreting into verbal language
The two numbers are 21 and ( 45 - 21 ) = 24
The two numbers are 21 and 24.

## Practice Problems

Problem 1: Solve for x: $\sqrt{2} - \sqrt{x + 6} = -\sqrt{x}$
Problem 2: Solve 12x - 6 = 6x + 12
Problem 3: Solve by elimination method:
x  - 7y  = 9
-2x + 3y  = 3
Problem 4: Solve system of linear equations: 3x + y = 5 and 2x - y = 0
Problem 5: Suresh is 4 times more aged than his daughter. After 6 years, Suresh's age will be 3 times of his daughter's age. Find their ages.