Lesson 2: Graphing Solutions of Two-Step Equations

Property

We can use graphs to find solutions to equations in one variable.

Examples

• To solve the equation $150 = 285 - 15x$ using the graph of $y = 285 - 15x$, find the point on the graph where the y-coordinate is 150. The x-coordinate of that point is $x=9$, which is the solution.

• To solve the inequality $285 - 15x \ge 150$ using the graph of $y = 285 - 15x$, find all points where the y-coordinate is 150 or more. The x-coordinates for these points are all values less than or equal to 9, so $x \le 9$.

Property

The graph of an equation in the form $y = mx + b$ is a vertical translation (or shift) of the graph of $y = mx$. The value of $b$ determines the direction and magnitude of the shift.

• If $b > 0$, the graph of $y = mx$ is shifted up by $b$ units.
• If $b < 0$, the graph of $y = mx$ is shifted down by $|b|$ units.

Examples

• The graph of $y = 2x + 5$ is the graph of $y = 2x$ shifted vertically upwards by 5 units.
• The graph of $y = -x - 3$ is the graph of $y = -x$ shifted vertically downwards by 3 units.
• The graph of $y = \frac{1}{2}x + 1$ is the graph of $y = \frac{1}{2}x$ shifted vertically upwards by 1 unit.

Explanation

This skill provides a way to visualize graphing two-step equations by transforming a simpler one-step equation. Since the line $y = mx$ always passes through the origin $(0, 0)$, the constant term $b$ shifts the y-intercept from $(0, 0)$ to $(0, b)$. For every x-value, the corresponding y-value on the line $y = mx + b$ is simply $b$ more than the y-value on the line $y = mx$. This results in the entire line moving up or down the y-axis without changing its steepness (slope).