In this chapter, we will learn to solve two linear equation using cartesian graph.
To understand the chapter, you should have basic understanding of linear equation and the method to plot it on the graph.
How to solve equations using graphs ?
Before learning to solve the equations, you have to understand what we are trying to do here.
We are finding common set of numbers that satisfy all the given equation.
This can be done by finding intersection point of two linear equation graph line.
The point at which two lines meet is the solution for given set of equation.
To solve system of linear equation by graphical method, follow the below steps;
(a) For the given linear equation, plot the line in graph.
(b) Locate the intersection of line and find its coordinates.
(c) The coordinates of intersected point is the solution of given equation.
I hope the above steps are clear. Now let us understand the method with help of examples.
Example 01
Solve the two linear equation using graphical method.
y = 2x + 1
y = x + 3
Solution
First plot y = 2x + 1 on graph.
Identify the two points that satisfy the above equation.
First Point
Put x = 0, we get;
y = 0 + 1
y = 1
We got the point (0, 1) that satisfy the equation.
Second Point
Put y = 0, we get;
0 = 2x + 1
2x = -1
x = -1/2
We got the point (-1/2, 0) from above calculation.
Using the points (0, 1) & (-1/2, 0), we can plot the graph for linear equation.
Now plot y = x + 3
Since the given expression is linear equation, it’s graph will be straight line.
We need two points to plot the given equation.
First Point
Put x = 0, we get;
y = 0 + 3
y = 3
So the point (0, 3) satisfy the equation y = x + 3.
Second Point
Put y = 0, we get;
0 = x + 3
x = -3
Hence, we get the point (-3, 0)
Now using the above two points (0, 3) & (-3, 0), we can draw the straight line.
In the above image;
Equation y = 2x + 1 is shown by black line.
Equation y = x + 3 is shown by red line
Both the lines intersect at point (2, 5), which is the solution of both the equation.
What does the solution (2, 5) signifies ?
It means that if you put the value (2, 5) in any of the two equation, it will be satisfied.
For example;
Put (2, 5) in equation y = 2x + 1
5 = 2 (2) + 1
5 = 4 + 1
5 = 5
Hence for (2, 5), the above equation is satisfied.
Similarly put (2, 5) in equation y = x + 3
5 = 2 + 3
5 = 5
Hence for (2, 5) the above equation satisfies.
So one can conclude that point (2, 5) is the solution of both the equations.
Example 02
Solve the below linear equation using graphical method.
2y = – x + 3
y = x + 6
Solution
Since both the given expression are linear equation, the graph of both the expression is straight line.
First plot equation 2y = – x + 3 in graphical form.
First Point
Put x = 0, we get;
2y = 0 + 3
y = 3 / 2
y = 1.5
Hence, we got the point (0, 1.5) which satisfy the above equation.
Second Point
Put y = 0, we get;
0 = -x + 3
x = 3
Hence we got the point (3, 0) as solution.
Now using the point (0, 1.5) & (3, 0), we can plot the straight line.
Now plot equation y = x + 6 in cartesian graph.
First Point
Put x = 0, we get;
y = 0 + 6
y = 6
We get the point (0, 6) that satisfy the given equation.
Second Point
Put y = 0, we get;
0 = x + 6
x = -6
We got the point (-6, 0) from above calculation.
Now plotting the graph using points (0, 6) and (-6, 0).
The red line in below image is the graphical representation of y = x + 6.
Note that both the lines intersect at point (-3, 3).
Hence, (-3, 3) is the solution of both the equation.
Example 03
Solve the below linear equation graphically.
2x – 4y = 8
4x – 8y = 20
Solution
Let’s plot the first equation; 2x – 4 y = 8
First Point
Put x = 0, we get;
0 – 4y = 8
y = -2
We get the point (0, -2) that satisfy the above equation.
Second Point
Put y = 0, we get;
2x – 0 = 8
x = 4
We get the point (4, 0) from above calculation.
We can plot the graph using the two points (0, -2) and (4, 0).
Now let’s plot the second equation, 4x – 8y = 20.
First Point
Put x = 0, we get;
0 – 8y = 20
y = -2.5
So we get the point (0, -2.5) that satisfy the above equation.
Second Point
Put y = 0, we get;
4x – 0 = 20
x = 5
So we get the point ( 5, 0 ) from above calculation.
Plotting the graph using above two points.
In the below image, the red line is graphical representation of equation 4x – 8y = 20
Note that both the lines are parallel and do not intersect each other.
Hence there is no solution for given algebraic expression.