# Solve Systems of Equations by Graphing

Linear equation is an algebraic equation which may consist of a single variable and constant or product of two or more constants. By term solving linear equation, we mean that we are going to find the value of the variable in the given equation. There are various methods to evaluate the system of linear equations, Graphing method is one of those.  Graphical representation is more meaningful and powerful way to communicate the information. Graphs provide visual and quick summaries of solution sets. In this section will learn how to design the graphs and find the set of solutions for the linear equations with 2 variables.

## How to Solve Linear Equation by Graphical Method

While solving system of linear equations in two variables, we need to remember the following points.
The graph of the linear equations is a straight line.
Consider a set of linear equations contains 2 variables x and y.
$a_1$ x + $b_1$y = $c_1$ and
$a_2$ x + $b_2$ y = $c_2$
The graph of the each equation is a straight line, the above pair of equations will satisfy any one of the following conditions.
(1) Lines intersect at a point.
(2) Lines are parallel
(3) the two lines coincide.

When we graph the lines on a coordinate plane, one of the following conditions will be satisfied.
1. If the lines intersect at a point there will be one and only one solution. The solution is written in the form (x,y).
2. If the lines are parallel there will be no solution, as the lines do not meet.
3. If the lines coincide there will be infinite number of solutions.

## Solving Systems of Equations by Graphing Examples

Let us see with the help of an example how to solve simultaneous linear equations using graphs.
Example: Solve 4x - 3y = 12 and x + y = 3

Solution
Find x and y intercepts of each equation and record it in the tables.
For 4x - 3y = 12
-3y = 12 - 4x
y = $\frac{12}{-3}$ - $\frac{4x}{-3}$
y = -4 +  $\frac{4x}{3}$
or y =  $\frac{4x}{3}$ - 4

 x 0 1 2 3 y -4 2.67 -1.33 0

For x + y = 3
y = 3 - x

 x 0 1 2 3 y 3 2 1 0

Plot a graph: Graph of 4x - 3y = 12 (Red line) and graph of x + y = 3 (Blue line)

Since point of intersection is the solution. The solution set for the given equations is (x, y) = (3, 0).

## Practice Problems

Practice below problems to sharp your skills in linear algebra.
Problem 1: Solve 2x + y = 9 and 3x - y = -4 using graphing method.
Problem 2: Find the solution set for system of linear equations using graphical method.
10x + 20 y = 30 and 2x - 4y = 10