101 LearnersLast updated on July 16th, 2025
Graphing Linear Equations
Linear equations are algebraic equations in which the highest degree of the variable is 1. Graphing a linear equation involves a visual representation of the equation on a graph. In this article, we will learn how to graph linear equations.
What is Graphing Linear Equations?
Graphing linear equations means plotting them on a coordinate plane, where every point on the line represents a solution to the equation. The linear equations have the highest degree of 1 and are written in the form y = mx + b, also known as the y-intercept form. The graph of a linear equation with one or two variables is always a straight line, and every point on the line represents a solution of the equation.
The point where the line crosses the x-axis is called the x-intercept, and the point where it crosses the y-axis is called y-intercept.
How to Represent Linear Equations Graphically?
Graphing a linear equation involves the process of finding its solutions and displaying them on the coordinate plane. Usually, two points (x, y) are used to plot the graph. Follow these steps to plot linear equations:
- Rewrite the equation in slope-intercept form, that is, y = mx + b.
- Choose simple x-values to find corresponding y-values to form coordinate pairs (x, y). The solution will be in the form (x, y).
- Find the value of x and y intercepts by substituting y = 0 and x = 0, respectively. The points will be (0, y) and (x, 0).
- Arrange the selected x values and their corresponding y values in a table.
- Plot the points on the graph and join all the points with a straight line to represent the linear equation.
Graphing Linear Equations in Two Variables
A linear equation in two variables is the equation in the form: ax + by = c or in the slope-intercept form (y = mx + b), where x and y are variables and a, b, and c are real numbers. To graph a linear equation in two variables, follow these steps:
- First, rewrite the equation in the form y = mx + b to identify the slope and the y-intercept
- Find two or three solutions by giving different values to x and find the corresponding values of y.
- Then find the x and y intercepts
- Plot the points on the graph, and connect them using a straight line.
Real-World Applications of Graphing Linear Equations
Learning to graph linear equations helps students to solve linear equations. It is used to track expenses, predict profits, and analyze scientific data. Let’s learn some applications of graphing linear equations.
- Graphing linear equations helps in budgeting and financial planning by visually tracking income, expenses, and savings over a period.
- In physics, linear equations are used to describe constant motion. For example, if the car goes at a speed of 50 km/h, the distance traveled can be modeled by y = 50x, where x is the time in hours and y is the distance in kilometers.
- In business, we use linear equations to model costs, such as fixed costs and variable costs, to predict profits.
Common Mistakes and How to Avoid Them in Graphing Linear Equations
Students often make mistakes while graphing linear equations. In this section, we will discuss some common mistakes and find ways to avoid them.
Mistake 1
Incorrectly identifying the slope
Students tend to misread the slope in y = mx + b. For example, in the equation y = 3x + 2, students often think the slope is 2 instead of 3. To avoid this error, always remember that in y = mx + b, m is the slope and b is the y-intercept.
Mistake 2
Misinterpreting equations in standard form
Students sometimes graph the linear equation in the standard form, Ax + By = C, but it can lead to errors, as they misidentify the slope or intercepts in the graph. For example, in the equation 2x + 3y = 6, students often misinterpret 2 as slope and 3 as y-intercept; it is wrong. So, always convert the equation in standard form to slope-intercept form before graphing it.
Mistake 3
Plotting the y-intercept on the x-axis
Plotting the y-intercept on the x-axis is a common mistake students make when plotting the linear equation. For example, in y = 2x + 3, students plot the point (3, 0) instead of (0, 3), thinking it is the y-intercept. To avoid this error, always remember that the y-intercept is the point where the graph crosses the y-axis and where the value of x is 0.
Mistake 4
Confusing x and y coordinates
When plotting the graph, students confuse x and y coordinates, for example, for points (-2, 4), students plot the graph with the point (-4, 2) instead of (-2, 4). To avoid this error, always remember that the first value (x) is the x-coordinate and the second value (y) is the y-coordinate.
Mistake 5
Not labeling the axes
When plotting the linear equations, students sometimes forget to label the axes. To avoid this, make sure to label both the axes (x-axis and y-axis) and scale on the graph
Solved Examples on Graphing Linear Equations
Problem 1
A line passes through the y-axis at 1 and has a slope of 2. What is the graph of this line?
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Explanation
Given, slope (m) = 2
y-intercept (b) = 1
In slope-intercept form, it can be written as y = 2x + 1
From y-intercept (0, 1), use slope 2 (go up 2, right 1), to get (1, 3).
Problem 2
Plot the graph of the equation, 2x + y = 8
na
Explanation
Given,
2x + y = 8
y = -2x + 8
Finding the value of y
x
y = -2x + 8
0
y = -2(0) + 8 = 8
2
y = -2(2) + 8 = 4
4
y = -2(4) + 8 = 0
Plot the points (0, 8), (2, 4), and (4, 0) and connect them.
Problem 3
Plot the graph of the equation, 15x - 5y = 25
na
Explanation
Finding the value of y to plot the graph,
15x - 5y = 25
-5y = 25 - 15x
Dividing the equation by -5:
y = -3x + 5
Finding the value of y
x
y = 3x - 5
0
y = 3(0) - 5 = -5
1
y = 3(1) - 5 = -2
2
y = 3(2) - 5 = 1
So, here the points are (0, -5), (1, -2), (2, 6)
Plotting the graph with these points.
Problem 4
Plot the graph of the following equation: x = 7,Plot the graph y = -2x
na
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Explanation
Here, the value of x is always 7; to plot x = 7, we draw a vertical line through (7, y).
Here, the points are (7, 0), (7, 2), (7, -3).
,In y = -2x, the slope is -2 and the y-intercept is 0. Finding the value of y, x y = -2x -1 2 0 0 1 -2 Here, the points are (-1, 2), (0, 0), (1, -2) Plotting the graph through the points
FAQs on Graphing Linear Equations
1.What is a linear equation?
2.What is the y-intercept?
3.What is the slope-intercept form?
4.Can a linear equation have a vertical or horizontal line?
5.How do you find the y-intercept of a graph?
6.How does learning Algebra help students in United Kingdom make better decisions in daily life?
7.How can cultural or local activities in United Kingdom support learning Algebra topics such as Graphing Linear Equations ?
8.How do technology and digital tools in United Kingdom support learning Algebra and Graphing Linear Equations ?
9.Does learning Algebra support future career opportunities for students in United Kingdom?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
