Graphing Two-Variable Equations

A two-variable equation, like $y = 2x + 1$, shows a mathematical relationship between two unknown values, usually $x$ and $y$. The graph of a linear two-variable equation is always a straight line on the coordinate plane. Every single point on that line represents a valid solution to the equation.

What is a Solution?

A solution to a two-variable equation is an ordered pair $(x, y)$ that makes the equation true when you plug the numbers in.

For example, let's check if $(3, 1)$ is a solution to the equation $2x - y = 5$:

1. Substitute $x = 3$ and $y = 1$ into the equation: $2(3) - 1 = 5$.
2. Simplify the math: $6 - 1 = 5$, which becomes $5 = 5$.

Since $5 = 5$ is a true statement, the ordered pair $(3, 1)$ is indeed a solution! If we graphed this equation, the line would pass exactly through the point $(3, 1)$.

3 Steps to Graph an Equation

To graph an equation, you need to find a few solutions, plot them as points, and connect them. Let's graph $y = 2x + 1$.

Step 1: Make a Table of Values Choose a few easy values for $x$ (like $0$, $1$, and $2$) and plug them into the equation to calculate the matching $y$ values.

• If $x = 0$, then $y = 2(0) + 1 = 1$. The ordered pair is $(0, 1)$.
• If $x = 1$, then $y = 2(1) + 1 = 3$. The ordered pair is $(1, 3)$.
• If $x = 2$, then $y = 2(2) + 1 = 5$. The ordered pair is $(2, 5)$.

Step 2: Plot the Points Draw an $x$-axis (horizontal) and a $y$-axis (vertical) to create a coordinate plane. Plot your three ordered pairs: $(0, 1)$, $(1, 3)$, and $(2, 5)$.

Step 3: Connect the Points Use a ruler or straightedge to draw a single line that passes through all your points. Add arrows to both ends of the line to show that the solutions continue infinitely in both directions.

Graphing Equations in Other Forms

Sometimes equations are written differently, such as $x + y = 5$. You can still use the exact same table method!

Find points by picking simple values for $x$:

• If $x = 0$, then $0 + y = 5 \implies y = 5$. Point: $(0, 5)$.
• If $x = 2$, then $2 + y = 5 \implies y = 3$. Point: $(2, 3)$.
• If $x = 5$, then $5 + y = 5 \implies y = 0$. Point: $(5, 0)$.

Plot the points $(0, 5)$, $(2, 3)$, and $(5, 0)$ on your graph, connect them with a straight line, and you have successfully graphed the equation!