PSAT 8/9 Math: Linear Equations in Two Variables
Linear equations in two variables describe straight lines, and the PSAT 8/9 tests how comfortable you are moving between a line's equation, its graph, and real-world meaning. This free, complete lesson covers slope-intercept form, parallel and perpendicular lines, intercepts, and interpreting linear models in context. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Slope-Intercept Form
A line's equation in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Given a slope and a point, substitute to solve for b, or read directly off a graph when the line is drawn.
Worked Example: Write the equation of a line with slope 3/4 that passes through (0, −2).
Since the point has x = 0, it's already the y-intercept: b = −2.
y = (3/4)x − 2
1. Write the equation of a line with slope 3/4 that passes through (0, −2). (see worked example above)
Explanation: Since the given point (0, −2) has x = 0, it's already the y-intercept, so b = −2, giving y = (3/4)x − 2.
2. A line has a slope of 2 and passes through (3, 11). Write its equation in slope-intercept form.
Explanation: Substitute (3, 11) into y = 2x + b: 11 = 2(3) + b → 11 = 6 + b → b = 5. So y = 2x + 5.
3. A table shows a line passing through (0, 8) and (4, 20). Write its equation in slope-intercept form.
Explanation: Slope = (20 − 8)/(4 − 0) = 12/4 = 3. The y-intercept is already given: (0, 8) means b = 8. So y = 3x + 8.
2. Parallel and Perpendicular Lines
Parallel lines have the exact same slope. Perpendicular lines have slopes that are negative reciprocals of each other (flip the fraction and change the sign). If a line is given in standard form, rearrange it into slope-intercept form first to identify its slope.
Worked Example: Line p is 4y − 8x = 16. Find the slope of a line perpendicular to p.
Rearrange: 4y = 8x + 16 → y = 2x + 4. So the slope of p is 2.
Perpendicular slope = negative reciprocal of 2 = −1/2
1. Line p is 4y − 8x = 16. Find the slope of a line perpendicular to p. (see worked example above)
Explanation: Rearranging 4y − 8x = 16 gives y = 2x + 4, so the slope of p is 2. A perpendicular slope is the negative reciprocal: −1/2.
2. A line is given by y = −5x + 7. Find the slope of a line parallel to it.
Explanation: Parallel lines share the exact same slope, so the parallel slope is also −5.
3. Line ℓ is 2x + 5y = 10. Find the slope of a line perpendicular to ℓ.
Explanation: Rearranging 2x + 5y = 10 gives y = (−2/5)x + 2, so the slope of ℓ is −2/5. The perpendicular slope is the negative reciprocal: 5/2.
3. Intercepts
The x-intercept is where a line crosses the x-axis (where y = 0), and the y-intercept is where it crosses the y-axis (where x = 0). Given both intercepts, you can find the slope directly: slope = (change in y) / (change in x) between the two intercept points.
Worked Example: A line has an x-intercept of (3, 0) and a y-intercept of (0, −6). Find its slope.
Slope = (−6 − 0) / (0 − 3) = −6 / −3 = 2
1. A line has an x-intercept of (3, 0) and a y-intercept of (0, −6). Find its slope. (see worked example above)
Explanation: Slope = (−6 − 0)/(0 − 3) = −6/−3 = 2.
2. A line has the equation y = 4x − 12. Find its x-intercept.
Explanation: Set y = 0: 0 = 4x − 12 → 12 = 4x → x = 3. The x-intercept is (3, 0).
3. A line passes through x-intercept (−4, 0) and y-intercept (0, 8). Write its equation in slope-intercept form.
Explanation: Slope = (8 − 0)/(0 − (−4)) = 8/4 = 2. The y-intercept is given directly: b = 8. So y = 2x + 8.
4. Interpreting Linear Models in Context
A two-variable linear equation can represent a real situation, where each coefficient has a specific meaning. Carefully match each term in the equation to what it represents in the scenario before answering.
Worked Example: The equation 2x + 5y = 40 models a lemonade stand, where x is the number of small cups poured and y is the number of large cups poured, using a fixed total amount of lemonade. What does the term 2x represent?
2x represents the total amount of lemonade used for small cups (2 units of lemonade per small cup, times the number of small cups).
1. The equation 2x + 5y = 40 models a lemonade stand, where x is the number of small cups poured and y is the number of large cups poured, using a fixed total amount of lemonade. What does the term 2x represent? (see worked example above)
Explanation: Since x counts small cups and each uses 2 units of lemonade, 2x is the total lemonade used for small cups.
2. The equation 3x + 12y = 2100 models trees planted, where x is trees planted per acre on a 3-acre field and y is trees planted per acre on a 12-acre forest, for a fixed total number of trees. What does y represent?
Explanation: Since 12y represents the total trees in the forest (12 acres times y trees per acre), y by itself is the rate: trees planted per acre in the forest.
3. A taxi charges a \$4 base fee plus \$2.50 per mile. Which equation models the total fare F for a trip of m miles?
Explanation: The per-mile rate ($2.50) multiplies the number of miles (m), and the one-time base fee ($4) is added separately: F = 2.5m + 4.
Common Mistakes to Avoid
- Forgetting to rearrange standard-form equations before comparing slopes. The coefficient of x in Ax + By = C is NOT the slope, you must solve for y first.
- Confusing parallel and perpendicular slope rules. Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals (flip and negate).
- Mixing up which coordinate is zero for each intercept. The x-intercept has y = 0, and the y-intercept has x = 0, it's easy to swap these by accident.
- Misreading which variable corresponds to which real-world quantity in a context problem, always re-read the problem's variable definitions carefully before interpreting a term.
- Taking the reciprocal without also flipping the sign (or vice versa) when finding a perpendicular slope, both steps are required.
Frequently Asked Questions
How do I quickly find a line's slope from standard form (Ax + By = C)?
You can rearrange into slope-intercept form, or use the shortcut: slope = −A/B directly from the standard-form coefficients.
Do parallel lines ever have the same y-intercept?
If two "parallel" lines have the same slope AND the same y-intercept, they're actually the same line, not two distinct parallel lines. True parallel lines share a slope but have different y-intercepts.
Where can I practice more problems like these?
The Linear Equations in Two Variables quizzes in the PSAT 8/9 Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT 8/9. You can also browse the full PSAT 8/9 Math Question Bank.