Lesson 4.2: Graph Linear Equations in Two Variables

New Concept

The graph of a linear equation is a straight line where every point represents a solution. We'll explore this relationship by plotting points to draw lines, including special cases like vertical ($x=a$) and horizontal ($y=b$) lines.

What’s next

Now, let's put this into practice. You'll begin with worked examples and then test your skills on a series of interactive practice cards.

Property

The graph of a linear equation $Ax + By = C$ is a line.

• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

Examples

• For the equation $y = 2x + 1$, the point $(2, 5)$ is a solution because $5 = 2(2) + 1$. Therefore, the point $(2, 5)$ is on the graph of the line.
• For the same equation, $y = 2x + 1$, the point $(3, 6)$ is not a solution because $6 \neq 2(3) + 1$. Therefore, the point $(3, 6)$ is not on the line.
• Every point on the graph of $x + y = 5$, such as $(1, 4)$ and $(5, 0)$, represents a pair of numbers that are solutions to the equation.

Property

To graph a linear equation by plotting points:
Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Examples

• To graph $y = x + 3$, find three solution points like $(0, 3)$, $(1, 4)$, and $(-1, 2)$. Plot these points and draw a line through them.
• To graph $y = \frac{1}{2}x - 1$, choose multiples of 2 for $x$ to avoid fractions. Good choices would be $(0, -1)$, $(2, 0)$, and $(4, 1)$.
• For $3x + y = 4$, first rewrite it as $y = -3x + 4$. Then find points by choosing values for $x$, such as $(0, 4)$, $(1, 1)$, and $(2, -2)$.

Explanation

This method is like an algebraic connect-the-dots. Find at least three coordinate pairs that solve the equation, place them on the graph, and draw a straight line through them. Using three points helps you catch any calculation mistakes.

Property

A vertical line is the graph of an equation of the form $x = a$. The line passes through the $x$-axis at $(a, 0)$. In this type of equation, the value of $x$ is always equal to $a$, no matter the value of $y$.

Examples

• The graph of the equation $x = 5$ is a vertical line where every point has an x-coordinate of 5, such as $(5, 0)$, $(5, 2)$, and $(5, -3)$.
• The equation $x = -2$ represents a vertical line that passes through the x-axis at the point $(-2, 0)$.
• A vertical line that passes through the point $(7, 4)$ has the equation $x = 7$ because the x-coordinate must always be 7.

Explanation

When an equation only has an $x$ variable, like $x = 2$, it means $x$ is always fixed at that number. No matter how high or low you go on the y-axis, the result is a perfectly straight, up-and-down vertical line.

Property

A horizontal line is the graph of an equation of the form $y = b$. The line passes through the $y$-axis at $(0, b)$. In this type of equation, the value of $y$ is always equal to $b$, no matter the value of $x$.

Examples

• The graph of the equation $y = 4$ is a horizontal line where every point has a y-coordinate of 4, such as $(0, 4)$, $(2, 4)$, and $(-1, 4)$.
• The equation $y = -1$ represents a horizontal line that passes through the y-axis at the point $(0, -1)$.
• A horizontal line that passes through the point $(5, -6)$ has the equation $y = -6$ because the y-coordinate must always be -6.

Explanation

When an equation only has a $y$ variable, like $y = 3$, it means $y$ is always fixed at that number. No matter how far left or right you move along the x-axis, the result is a perfectly flat, horizontal line.

Property

An equation with both $x$ and $y$ (like $y = 4x$) shows that the value of $y$ depends on the value of $x$. This dependency creates a slanted line.
An equation with only one variable (like $y = 4$) has a constant value, which creates a horizontal or vertical line.

Examples

• The graph of $y = 5x$ is a slanted line because the value of $y$ changes with $x$. The graph of $y = 5$ is a horizontal line because $y$ is always 5.
• The equation $y = -x$ produces a slanted line. For instance, the points $(1, -1)$ and $(2, -2)$ are both solutions on this line.
• The equation $y = -3$ produces a horizontal line. For instance, the points $(0, -3)$ and $(5, -3)$ are both solutions on this line.

Explanation

An equation like $y = 2x$ describes a relationship where $y$ changes as $x$ changes, creating a slope. In contrast, an equation like $y = 2$ is a rule where $y$ is always 2, creating a flat, horizontal line.