Graphing Linear Equations

A linear equation in two variables produces a straight line when graphed on a coordinate plane. Learning to graph these equations lets you visualize relationships between quantities — costs and quantities, dosages and weights, distance and time — and quickly identify trends, intercepts, and break-even points.

Slope-Intercept Form: $y = mx + b$

This is the most common form for graphing. The equation gives you two pieces of information directly:

• $m$ = the slope (steepness and direction of the line)
• $b$ = the y-intercept (where the line crosses the $y$-axis)

How to Graph from Slope-Intercept Form

1. Plot the y-intercept $(0, b)$ on the $y$-axis
2. Use the slope to find a second point: from the y-intercept, move up (or down) by the rise and right by the run
3. Draw a straight line through both points and extend it in both directions

Example 1: Graph $y = 2x + 1$

Step 1 — Identify slope and y-intercept:

$m = 2 = \frac{2}{1}$ (rise 2, run 1), $b = 1$

Step 2 — Plot the y-intercept: Place a point at $(0, 1)$.

Step 3 — Use the slope: From $(0, 1)$, move up 2 and right 1 to reach $(1, 3)$.

Step 4 — Plot a third point for accuracy: From $(1, 3)$, move up 2 and right 1 to reach $(2, 5)$.

Step 5 — Draw the line through $(0, 1)$, $(1, 3)$, and $(2, 5)$.

Example 2: Graph $y = -\frac{3}{4}x + 6$

Step 1 — Identify slope and y-intercept:

$m = -\frac{3}{4}$ (down 3, right 4), $b = 6$

Step 2 — Plot the y-intercept: Place a point at $(0, 6)$.

Step 3 — Use the slope: From $(0, 6)$, move down 3 and right 4 to reach $(4, 3)$.

Step 4 — Draw the line through the points. The negative slope means the line falls from left to right.

Graphing Using X and Y Intercepts

An alternative method that works well when the equation is not in slope-intercept form.

• The y-intercept is found by setting $x = 0$
• The x-intercept is found by setting $y = 0$

Example 3: Graph $3x + 2y = 12$

Find the y-intercept (set $x = 0$):

$3(0) + 2y = 12 \implies 2y = 12 \implies y = 6$

Y-intercept: $(0, 6)$

Find the x-intercept (set $y = 0$):

$3x + 2(0) = 12 \implies 3x = 12 \implies x = 4$

X-intercept: $(4, 0)$

Draw the line through $(0, 6)$ and $(4, 0)$.

Point-Slope Form: $y - y_1 = m(x - x_1)$

Use this form when you know the slope and one point on the line (but not necessarily the y-intercept).

Example 4: Write and graph the equation of a line with slope $3$ passing through $(2, 5)$

Step 1 — Substitute into point-slope form:

$y - 5 = 3(x - 2)$

Step 2 — Convert to slope-intercept form (optional, for graphing):

$y - 5 = 3x - 6$

$y = 3x - 1$

Step 3 — Graph: The y-intercept is $(0, -1)$ and the slope is 3. Plot $(0, -1)$, then move up 3 and right 1 to $(1, 2)$, then to $(2, 5)$.

Horizontal and Vertical Lines

Horizontal lines have the form $y = c$ (where $c$ is a constant). The slope is 0.

• Example: $y = 4$ is a flat line crossing the $y$-axis at 4

Vertical lines have the form $x = c$. The slope is undefined.

• Example: $x = -2$ is a vertical line crossing the $x$-axis at $-2$

Vertical lines are not functions because a single $x$-value maps to infinitely many $y$-values.

Real-World Application: Retail — Break-Even Analysis

A small business sells custom phone cases. The costs and revenue can be modeled as linear equations:

• Cost: $y = 5x + 200$ (each case costs $5 to produce, plus $200 in fixed monthly expenses)
• Revenue: $y = 12x$ (each case sells for $12)

To graph these:

Cost line: y-intercept at $(0, 200)$, slope of 5 (up 5, right 1)

Revenue line: y-intercept at $(0, 0)$, slope of 12 (up 12, right 1)

The break-even point is where the two lines intersect. Setting cost equal to revenue:

$5x + 200 = 12x$

$200 = 7x$

$x \approx 28.6$

The business must sell approximately 29 cases to break even. On a graph, you can see this visually: below 29 cases the cost line is above the revenue line (operating at a loss), and above 29 cases the revenue line is higher (profit).

Common Mistakes to Avoid

1. Plotting slope backwards. Slope is $\frac{\text{rise}}{\text{run}}$, not $\frac{\text{run}}{\text{rise}}$. For $m = \frac{2}{3}$, move up 2 and right 3 — not up 3 and right 2.

2. Forgetting the negative in a negative slope. For $m = -\frac{3}{4}$, you move down 3 and right 4 (or up 3 and left 4).

3. Confusing x-intercept and y-intercept. The y-intercept is where $x = 0$; the x-intercept is where $y = 0$.

4. Not plotting enough points. Two points define a line, but plotting a third point helps you catch errors.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Slope-intercept form ($y = mx + b$) gives you the slope and y-intercept directly — it is the fastest way to graph a line
• Point-slope form ($y - y_1 = m(x - x_1)$) is useful when you know a point and the slope but not the y-intercept
• Intercept method: set $x = 0$ to find the y-intercept and $y = 0$ to find the x-intercept, then connect them
• Horizontal lines ($y = c$) have zero slope; vertical lines ($x = c$) have undefined slope
• Graphing lets you visualize relationships and quickly identify key values like break-even points and trends

Return to Algebra for more topics in this section.