In this chapter we will learn to graph linear functions or linear equations in cartesian coordinates.
Before moving in to core topic, let us first review the basics of linear equations.
What are Linear Equation ?
The linear equations is a equation for straight line.
In linear equations we have two variables and a constant.
Generally the linear equation is expressed in following ways;
(a) General Representation
Ax + By = C
In the above equation;
x & y ⟹ are the variables
A ⟹ coefficient of x
B ⟹ coefficient of y
C ⟹ constant value C
(b) Slope Intercept form
y = mx + c
In the above expression;
y & x ⟹ are the variables
m ⟹ slope of equation line
c ⟹ intercept formed by equation line
The slope intercept form is convenient expression to graph any given linear equation.
Note:
In linear equation, the power of variables is always 1.
If any of the expressions have power of 2 or more than the given equation is not linear.
Plotting Linear equation in cartesian graph
Understand that the linear equation is always a straight line and to graph a straight line, we need two points to join.
In the method to graph linear equation, we will try to extract two point using which we can draw straight line.
Given below are the steps to graph linear equation;
(a) Convert linear equation into slope intercept form.
(b) Find two set of points which can satisfy the equation.
This cab be done by using two methods;
(i) Put values x = 0 & y = 0 in the equation and get the values of respective points.
(ii) Use trial & error method to get the values.
(c) After getting the two points, plot them in cartesian graph.
(d) Join the two points with straight lines and you will get the linear equation line.
I hope you understood the above process. Let us solve some examples for better understanding.
Graphing Linear Equation – Solved Problems
Example 01
Graph the equation 6x + 2y = 8.
Solution
Given above is the linear equation because all the variables have highest power 1.
To graph the linear equation, follow the below steps;
(a) Convert the equation into slope intercept form ,y = mx + c
6x + 2y = 8
2y = -6x + 8
y = (-6x / 2) + (8 / 2)
y = -3x + 4
Hence, we get the linear equation y = -3x + 4 in slope intercept form.
(b) Get two set of points that satisfies linear equation.
First Point
Put x = 0 in equation
y = -3x + 4
y = -3(0) + 4
y = 4
Hence for x = 0, the value of y is 4.
The point (0, 4) will satisfy the given equation.
Second Point
Put y = 0 in equation.
y = -3x + 4
0 = -3x + 4
3x = 4
x = 4/3 = 1.33
Hence for y = 0, we get x = 1.33
Hence point (4/3, 0 ) will satisfy the given equation.
(c) Now plot points (0, 4) & (4/3, 0) on the graph & join them using straight line. The straight line formed is plot for our equation.
Example 02
Graph the equation, y = 2 + x
Solution
Follow the below steps to graph the above expression.
(a) Convert the equation into slope intersect form.
y = 2 + x
y = x + 2
(b) Find two points that satisfy the given equation.
First Point
Put x = 0 in the equation.
y = x + 2
y = 0 + 2
y = 2
For x= 0, we get the value y = 2.
Hence the point (0, 2) will satisfy the equation.
Second Point
Put y = 0 in the equation
y = x + 2
0 = x + 2
x = -2
For y = 0, we get x = -2.
Hence the point (-2, 0) will satisfy the give equation.
(c) Plot the points (0, 2) & (-2, 0) on graph and join them with straight lines.
The above straight line represents the graph of linear equation y = 2 + x
Example 03
Graph the linear equation y = 6x
Solution
Follow the below steps.
(a) Convert the linear equation into slope intercept form.
The equation is already in slope intercept form, y = mx + c
y = 6x + 0
y = 6x
(b) Find two points that satisfy the given equation.
First Point
Put x = 0 in the equation
y = 6x
y = 6 (0)
y = 0
So for x = 0, we get y = 0.
Hence the point (0, 0) satisfy the given equation.
Second Point
If we put y = 0, we will get x = 0.
So the point (0, 0) will be repeated again.
Here put x = 1;
y = 6x
y = 6(1)
y = 6
Hence, the points (1, 6) will satisfy the given equation.
(c) Now plot the two points (0, 0) and (1, 6) in the graph and join them with straight line.
The above line represents the linear equation y = 6x.
Example 04
Graph the linear equation, y – 2x = 4
Solution
Follow the below steps.
(a) Convert the equation into slope intercept form y = mx + c
y – 2x = 4
y = 2x + 4
(b) Find two points that satisfy the given equation.
First Point
Put x = 0 in the equation.
y = 2x + 4
y = 2 (0) + 4
y = 4
Hence, the point (0, 4) satisfy the equation.
Second Point
Put y = 0 in the equation.
y = 2x + 4
0 = 2x + 4
2x = -4
x = -2
Hence, the point (-2, 0) satisfy the given equation
(c) Plot the points (0, 4) and (-2, 0) on the graph and join them with straight line.
The above straight line represent the given linear equation y – 2x = 4.