SAT — mathMedium6 min read

Graphing Linear Equations

Learn to graph linear equations for the Digital SAT. Plot lines using slope and intercept, identify equations from graphs, and interpret key features.

Tutor AI Team2026-02-25T01:10:50.877Z

Graphing linear equations is a visual skill that brings algebra to life. On the Digital SAT, you will encounter questions that ask you to identify the equation of a graphed line, match an equation to a graph, or determine key features (slope, intercepts) from a graph. Understanding the connection between an equation and its graph is fundamental to success on the SAT Math section.

Core Concepts

Plotting from Slope-Intercept Form

Given $y = mx + b$ :

Plot the y-intercept $(0, b)$ on the y-axis.
Use the slope $m = \frac{\text{rise}}{\text{run}}$ to find a second point.
Draw the line through both points.

Example: Graph $y = 2x - 3$ .

y-intercept: $(0, -3)$
Slope: $\frac{2}{1}$ → from $(0, -3)$ , go up 2, right 1 → $(1, -1)$
Draw the line through $(0, -3)$ and $(1, -1)$ .

Positive vs. Negative Slope

Positive slope ( $m > 0$ ): line rises from left to right ↗
Negative slope ( $m < 0$ ): line falls from left to right ↘
Zero slope ( $m = 0$ ): horizontal line →
Undefined slope: vertical line ↑ (equation: $x = k$ )

Identifying the Equation from a Graph

To find the equation of a graphed line:

Read the y-intercept from the graph (where the line crosses the y-axis).
Calculate the slope by picking two clear points: $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Write the equation: $y = mx + b$ .

x-intercept and y-intercept

y-intercept: the point where the line crosses the y-axis (set $x = 0$ )
x-intercept: the point where the line crosses the x-axis (set $y = 0$ )

Example: For $y = 3x - 6$ :

y-intercept: $(0, -6)$
x-intercept: set $y = 0$ → $0 = 3x - 6$ → $x = 2$ → $(2, 0)$

Plotting from Standard Form

For $Ax + By = C$ , the easiest method is to find both intercepts:

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

x-intercept: $(6, 0)$
y-intercept: $(0, 4)$
Plot both points and draw the line.

Horizontal and Vertical Lines

Horizontal line: $y = k$ (slope = 0, every point has the same y-value)
Vertical line: $x = k$ (undefined slope, every point has the same x-value)

Strategy Tips

Tip 1: Use Intercepts for Quick Graphing

Finding the x- and y-intercepts gives you two points quickly, which is all you need to draw a line.

Tip 2: Estimate Slope from a Graph

On SAT graphs, pick two points that clearly fall on grid intersections. Calculate rise/run between them.

Tip 3: Eliminate Wrong Answers by Slope Sign

If the graphed line goes down from left to right, the slope must be negative. Eliminate any answer choice with a positive slope.

Tip 4: Check with a Point

If you identify a point on the graph, substitute it into each answer choice. The correct equation must be satisfied.

Tip 5: Steepness = |Slope|

A steeper line has a larger absolute value of slope. Use this to distinguish between similar answer choices.

Worked Example: Example 1

Problem

A line passes through $(0, 4)$ and $(2, 0)$ . What is its equation?

Slope: $m = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2$

y-intercept: $4$

Equation: $y = -2x + 4$

Solution

Worked Example: Example 2

Problem

Which of the following could be the equation of a line with positive slope and negative y-intercept?

A) $y = -3x + 2$ — negative slope ✗ B) $y = 2x - 5$ — positive slope, negative y-intercept ✓ C) $y = 2x + 5$ — positive slope, positive y-intercept ✗ D) $y = -x - 3$ — negative slope ✗

Answer: B

Solution

Worked Example: Example 3

Problem

The line $4x - 2y = 8$ is graphed in the xy-plane. What is the slope?

Convert: $-2y = -4x + 8$ → $y = 2x - 4$

Slope $= 2$

Solution

Worked Example: Example 4

Problem

A graph shows a line crossing the y-axis at 3 and passing through $(4, 5)$ . Find the equation.

y-intercept: $b = 3$

Slope: $m = \frac{5 - 3}{4 - 0} = \frac{2}{4} = \frac{1}{2}$

Equation: $y = \frac{1}{2}x + 3$

Solution

Worked Example: Example 5

Problem

Where does the line $y = -3x + 9$ cross the x-axis?

Set $y = 0$ : $0 = -3x + 9$ → $3x = 9$ → $x = 3$

x-intercept: $(3, 0)$

Solution

Practice Problems

Problem 1

A line has slope $-\frac{3}{2}$ and y-intercept $5$ . Write its equation.

Problem 2

Find the x- and y-intercepts of $5x + 2y = 20$ .

Problem 3

A graph shows a horizontal line passing through $(3, -2)$ . What is the equation?

Problem 4

A line passes through $(-1, 4)$ and $(3, -4)$ . What is the slope and equation?

Problem 5

Which line is steeper: $y = 3x + 1$ or $y = -5x + 2$ ?

Problem 6

A line crosses the x-axis at $(4, 0)$ and the y-axis at $(0, -6)$ . Write its equation in standard form.

Want to check your answers and get step-by-step solutions?

Common Mistakes

Reversing rise and run. Slope = rise/run = (change in y)/(change in x). Don't flip it.
Sign error in slope. If both coordinates change, track signs carefully: going down is negative rise.
Confusing x-intercept and y-intercept. The y-intercept is on the y-axis ( $x = 0$ ); the x-intercept is on the x-axis ( $y = 0$ ).
Misreading the scale on SAT graphs. Check whether grid lines represent 1 unit, 2 units, etc.
Plotting slope in the wrong direction. A slope of $-\frac{3}{2}$ means down 3, right 2 (or equivalently, up 3, left 2).

Frequently Asked Questions

Do I need to draw graphs on the SAT?

Not usually, but sketching a quick graph on your scratch paper can help you understand the problem and eliminate answers.

How precise do my graphs need to be?

For SAT purposes, a rough sketch is fine. You need to get the slope direction and intercept right, not pixel-perfect accuracy.

Can a line have a slope of 0?

Yes — that's a horizontal line, like $y = 3$ .

What's the slope of $x = 5$?

Undefined. Vertical lines have no defined slope.

How do I graph on the calculator?

On the SAT's built-in Desmos calculator, type the equation (e.g., $y = 2x + 1$ ) and it graphs automatically.

Key Takeaways

✓
y = mx + b: plot the y-intercept, then use slope to find more points.
✓
Positive slope → line rises; negative slope → line falls.
✓
To identify an equation from a graph: find the y-intercept and calculate the slope from two clear points.
✓
x-intercept: set $y = 0$ . y-intercept: set $x = 0$ .
✓
Use the Desmos calculator on the Digital SAT to verify graphs quickly.
✓
Eliminate wrong answers by checking slope sign and intercept values.

Tags:

#sat#math#algebra#graphing#linear-equations

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Core Concepts

Plotting from Slope-Intercept Form

Positive vs. Negative Slope

Identifying the Equation from a Graph

x-intercept and y-intercept

Plotting from Standard Form

Horizontal and Vertical Lines

Strategy Tips

Tip 1: Use Intercepts for Quick Graphing

Tip 2: Estimate Slope from a Graph

Tip 3: Eliminate Wrong Answers by Slope Sign

Tip 4: Check with a Point

Tip 5: Steepness = |Slope|

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: Example 4

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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