A linear equation is one of the first big hurdles in algebra, but graphing it is much simpler than it looks. Whether you are helping your child with homework or trying to make sense of coordinate grids yourself, turning an equation into a straight line comes down to a few repeatable steps.
In this guide, our Mathnasium tutors explain what linear equations mean, how slope and coordinates work together, and how to plot any equation step by step using a simple table of values.
A linear equation is an algebraic equation where each variable has an exponent of 1, and its graph on a coordinate plane forms a straight line.
Let’s see how this works through a simple example. Say we start a hike already 3 miles in, and we take a little break, and then we decide to walk at a steady pace of 2 miles every hour.
After one hour, we're at 5 miles.
After two hours, we're at 7 miles.
After three hours, we're at 9 miles.
Because the distance increases by the exact same amount every hour, the connection between time and distance stays the same, which makes this a linear relationship.
We can write that same relationship as an equation, where:
y represents the total distance covered
x represents the number of hours we've been walking, and multiplying it by our rate of 2 miles per hour tells us how far we've traveled since we started
The 3 miles is where we started, before any walking hours are counted, and that makes it our constant
Together, those pieces give us:
y = 2x + 3
We call this the slope-intercept form, and it’s one of the three common forms a linear equation can take.
|
Form: |
Equation: |
Best For: |
|
Slope-Intercept Form |
y = mx + b (m is the slope, b is the y-intercept) |
We know the slope and where the line crosses the y-axis. |
|
Standard Form |
Ax + By = C (A, B, and C are constants) |
We want both variables on the same side, useful for comparing equations. |
|
Point-Slope Form |
y − y₁ = m(x − x₁) |
We know the slope and one specific point on the line. |
Note: In each of these forms, x and y carry an exponent of 1. A higher power on x, such as x², produces a curve instead of a straight line, making the equation nonlinear.
The key properties of a linear equation graph are its slope and its x- and y-coordinates.
The slope determines how steeply the line rises or falls, and the x- and y-coordinates define the exact position of every point along the line. Together, these properties show us how the variables interact across the coordinate plane.
The slope tells us how steep a line is, and whether it climbs or falls. In y = mx + b, the slope is m, the number that multiplies x.
The value of m tells us how the line behaves:
Increasing if m > 0 (the line goes up)
Horizontal if m = 0 (the line is flat)
Decreasing if m < 0 (the line goes down)
If m = 0, the equation becomes y = 0x + b, which simplifies to y = b. With no rise built into the equation, the line stays flat at height b across the entire graph.
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Every point on the coordinate plane is written as (x, y), with the x-coordinate first and the y-coordinate second.
The x-coordinate tells us how far to move left or right
The y-coordinate tells us how far to move up or down
In y = mx + b, x is the value we choose, and y is what the equation gives us back once we plug that x value in.
Every pair of x and y that satisfies the equation becomes one point (x, y) we can plot on the line.
Take our earlier equation, y = 2x + 3. If we choose x = 1, the equation gives us y = 2(1) + 3 = 5.
That places our point at (1, 5) on the coordinate plane.
And we can plot it on the graph like this:
For x = 1, we move 1 unit to the right of the origin along the x-axis
For y = 5, we move 5 units up from there along the y-axis
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To graph a linear equation using a table of values, we choose a few x values, solve for the matching y value each time, and plot the resulting points.
Let’s put this into practice with y = -3x + 4.
Three values are usually enough, and it helps to include 0 as one of them. If x equals 0, the equation gives us the y-intercept directly, so that point is a useful anchor before we add the other two.
Let's choose x = -1, x = 0, and x = 1.
We replace x in y = -3x + 4 with each value we chose, then solve for y.
x = -1: y = -3(-1) + 4 = 7
x = 0: y = -3(0) + 4 = 4
x = 1: y = -3(1) + 4 = 1
Each x value and its matching y value form one point, written as (x, y). A table keeps all three pairs organized before we plot them.
|
x |
y |
|
-1 |
7 |
|
0 |
4 |
|
1 |
1 |
We locate every point by moving along the x-axis first, then up or down to match the y-coordinate.
For (-1, 7), we move 1 unit left of the origin, then 7 units up
For (0, 4), we stay at the origin on the x-axis, then move 4 units up
For (1, 1), we move 1 unit right of the origin, then 1 unit up
We draw a straight line through all three points, extending it in both directions with arrows on each end. Since all three points sit on the same line, this confirms that the equation is linear.
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Let's put the table of values method to work. For each equation, follow the same five steps we just covered to draw the line.
Linear Equation 1: y = 3x - 2
Linear Equation 2: y = -2x + 6
Take your time working through each one. Check your answers at the bottom of the guide.
A Mathnasium tutor helps a student build confidence in algebra, one concept at a time.
Mathnasium is a math-only learning center serving students, focused on building confident, independent math thinkers through personalized instruction and targeted support.
Students come to us at different points in their math journey. Some need extra support with core skills like plotting coordinates and reading a graph. Others are ready to work with slope and multiple equation forms at a more advanced level. Wherever a student is starting from, we build a path forward from that exact point.
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Teaching for Understanding: Our specially trained tutors use plain, everyday language and a mix of verbal, visual, mental, tactile, and written techniques so concepts like slope and coordinates land in a way that makes sense to each student.
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Here are the worked-out solutions for both practice equations. It doesn't matter which x values you chose. Every correct set of points lands on the same line, so your graph should match the ones here.
Solution 1: y = 3x - 2
Solution 2: y = -2x + 6
Mathnasium of Johns Creek is a math-only learning center for K-12 students in Johns Creek, GA. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
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