Equations are used everywhere in mathematics. An equation is defined to an expression with equal $(=)$ sign. Just to recall that expression is a combination of variables and constants. There are different types of equations in mathematics. Linear equations are studied in middle-level mathematics. There may be a set of linear equations each equation of which shares the same values of variables. Such set is known as a system of linear equations. Math is one subject that students find tough to understand. The difficult problems, endless practice questions and equations certainly don't make it easy. However, learning math can be made fun and less stressful if students are allowed to learn at their own pace. In this section you will be able to practice some linear equations problems designed by subject experts. Answers have been structured in a logical and easy language for quick revisions.

Examples on Linear Equations

We can directly solve equations with one variable using one step or two step solutions. Whereas to solve equations with two or more variables, we use listed methods: Substitution method, graphical method, elimination method etc. Let us solve some of the problems using these methods.

Example 1: Solve the set of simultaneous equation by graphing

$2x - y$ = $3$ and $x + y$ = $5$

Solution:

Rewrite both equations into slope intercept form.

$2x - y$ = $3$ can be written as $y$ = $2x - 3$

$x + y$ = $6$ can be written as $y$ = $-x + 6$

Graph both the lines on a same graph and if they intersect find the point of intersection and this point would be the solution.

Linear Equations Examples

$(3, 3)$ is the solution it means $x$ = $3,\ y$ = $3$.

One can cross check by plugging in the value into the original equations.

$2x - y$ = $3$ we can plug in $x$ = $3$ and $y$ = $3$ we get $2(3) - 3$ = $3$ = $6 - 3$ = $3$ is true, similarly
the second equation can be checked x + y = 6 plug in $x$ = $3$ and $y$ = $3$. $3 + 3$ = $6$.  Which is true. Our solution is right.

Example 2: Find the solution(s) common( if any )  to the given equations 3x + y = 10 , 6x + 2y = -5
Solution: Let us solve the given equations by elimination method.
Let us first number the equations. 
3x + y = 10  ---------------------(1)
6x + 2y = -5  ---------------------(2)
                                                         
Multiplying Equation (1) by 2, we get 2 ( 3x + y ) = 2 ( 10)
 6x  +  2y   = 20  ----------------------(3)
6x + 2y     = -5  -----------------------(2)
(-)  (-)       (+)
 ________________________
Subtracting (2) from (3), we get   
0  =  25 , which is  a false statement.
Hence the above system of equations have no solution.

Example 3: Solve the set of linear equation by Elimination Method

$3x + y$ = $5$

$2x - y$ = $0$

Solution

What we see one of variable in one equation is having the opposite sign in the other equation, in such cases we just add the two equations.

$3x + y$ = $5$
$2x - y$ = $0$
--------------------
$5x$ + 0y = $5$

This implies 5x = 5
or x = 1

Substitute the value of $x$ into any one of the equation and solve for $y$

$3x + y$ = $5$

$3(1) + y$ = $5$

$3 + y$ = $5$

$y$ = $2$

The solution is $x$ = $1,\ y$ = $2$, we can also represent it as an ordered pair $(1, 2)$

Practice Problems

Below listed problems are based on NCERT book. Practice-cum NCERT problems series for class 6 to 12 emphasizes that learning should be activity-based and knowledge driven.

Problem 1: Solve $\sqrt{3} - \sqrt{2x -1} = \sqrt{x}$
Problem 2: Solve 10x - 6 + 5x = 54x - 21
Problem 3: Solve by graphical method
y - 2x = 3
y = -x + 6
Problem 4: Solve pair of linear equations: -x + 2y = 3 and 12x - 6y = 10
Problem 5: The sum of 2 numbers is 105 and their ratio is 2:3. What are the numbers?