ORIGINAL WORKED EXAMPLE
Slope between unequal steps
Find the slope through (−1,8) and (3,2).
- The horizontal change is 3 − (−1) = 4.
- The vertical change is 2 − 8 = −6.
- Divide changes: m = −6/4 = −3/2.
Answer −3/2
SAT · Algebra · TOPIC LESSON
Linear equations in two variables
Read a line as all ordered pairs satisfying one relationship.
BY THE END, YOU CAN
- Translate among standard, slope-intercept and point-slope forms.
- Calculate slope and intercepts and interpret their units.
- Recognize parallel, perpendicular, horizontal and vertical lines.
BEFORE YOU START
- One-variable equation solving.
- Ordered pairs and coordinate axes.
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original SAT-style worked examples and practice
The teaching passages, worked examples and exercises in this lesson are original. They are not official College Board questions or past-exam reproductions. Blueprint and observed-practice analytics describe separate factual metadata; those source questions are not displayed here.
01 / UNDERSTAND & APPLY
An equation describes many pairs
A linear equation in two variables usually gives infinitely many ordered-pair solutions. In Ax + By = C, A and B are not both zero. An ordered pair (p,q) lies on its graph exactly when Ap + Bq = C. One equation does not normally determine both coordinates uniquely.
Coordinates have an order: substitute the first value for x and the second for y. In an application, a point may satisfy the algebraic equation but be excluded because a count cannot be negative or fractional.
02 / UNDERSTAND & APPLY
Slope measures change, not height
Slope m = (y₂ − y₁)/(x₂ − x₁), provided x₂ ≠ x₁. Use the same subtraction order in numerator and denominator. Its units are y-units per x-unit. A negative slope means y decreases as x increases; it does not mean all y-values are negative.
03 / UNDERSTAND & APPLY
Choose the form that exposes the information
Slope-intercept form y = mx + b exposes slope m and y-intercept b. Point-slope form y − y₀ = m(x − x₀) uses one known point. Standard form Ax + By = C often arises naturally from totals. Equivalent equations can be multiplied by any nonzero constant.
When B ≠ 0, rearranging standard form gives y = (−A/B)x + C/B. Thus the coefficient A alone is not the slope.
ORIGINAL WORKED EXAMPLE
Build a line from a point and slope
Find an equation of the line with slope 3 through (2,−1).
- Use point-slope form: y − (−1) = 3(x − 2).
- Distribute: y + 1 = 3x − 6.
- Subtract 1: y = 3x − 7.
- Check the point: 3(2) − 7 = −1.
Answer y = 3x − 7
04 / UNDERSTAND & APPLY
Intercepts are zero-coordinate cases
At an x-intercept, y = 0; at a y-intercept, x = 0. An intercept is a point, although a problem may ask only for its nonzero coordinate. An axis crossing can have contextual meaning only when zero input or zero output is meaningful.
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ORIGINAL WORKED EXAMPLE
A resource constraint
A workshop has 24 hours. Each large item needs 4 hours and each small item needs 3 hours. For 4x + 3y = 24, interpret both intercepts.
- Set y = 0: 4x = 24, so the x-intercept is (6,0).
- Set x = 0: 3y = 24, so the y-intercept is (0,8).
- The workshop can make 6 large items and no small items, or 8 small items and no large items, using all 24 hours.
Answer (6,0) and (0,8)
05 / UNDERSTAND & APPLY
Direction relationships
Distinct nonvertical parallel lines have equal slopes and different y-intercepts. Nonvertical perpendicular lines with defined, nonzero slopes satisfy m₁m₂ = −1. A vertical line is perpendicular to a horizontal line; that case is better recognized geometrically than forced into a slope formula.
Equal slopes and equal intercepts describe the same line, not two distinct parallel lines.
ORIGINAL WORKED EXAMPLE
A perpendicular line
Find the line through (4,1) perpendicular to y = 2x + 5.
- The given slope is 2. The perpendicular slope is −1/2 because 2(−1/2) = −1.
- Use the point: y − 1 = −(1/2)(x − 4).
- Simplify: y = −x/2 + 3.
Answer y = −x/2 + 3
06 / UNDERSTAND & APPLY
Vertical and horizontal are different
The equation y = k is horizontal: all x-values are allowed and the slope is zero. The equation x = h is vertical: x is fixed and the slope is undefined because the horizontal change is zero. A vertical line cannot be written as y = mx + b with a real slope.
ORIGINAL WORKED EXAMPLE
A zero denominator is information
What is the line through (−3,2) and (−3,7)?
- Both points have x = −3, while y changes.
- The slope ratio is (7 − 2)/(−3 − (−3)) = 5/0, which is undefined.
- The equation is x = −3, not y = −3.
Answer x = −3
07 / UNDERSTAND & APPLY
Coefficients in a real model
In y = 84 − 6x, if y is liters remaining after x minutes, the initial amount is 84 liters and the rate is −6 liters per minute. The x-intercept is 14 minutes, but after that the model predicts negative volume and must stop. A mathematical line extends forever; a contextual model need not.
To convert a per-minute slope to a per-hour slope, multiply by 60 because an hour contains 60 minutes. Changing units changes the numerical slope without changing the underlying relationship.
08 / UNDERSTAND & APPLY
A table, an equation and a constraint
For the workshop equation 4x + 3y = 24, the table of pairs (0,8), (3,4), (6,0) describes three possible complete uses of the time. Substituting x = 3 gives 12 + 3y = 24, hence y = 4. The term 4x is hours spent on large items, not the number of large items.
To recover an equation from a linear table, find the slope from any two distinct x-values, use one point to find b, and check the remaining rows. For example, pairs (1,7), (3,3), (4,1) give slope −2 and y = −2x + 9. Graph these points and draw the line only when a continuous linear relationship is intended. Integer production counts are isolated feasible points on the line.
09 / UNDERSTAND & APPLY
Check more than appearance
Substitute a known point and compare the slope to verify an equation. A calculator graph helps connect forms, but visual steepness depends on axis scale. Read the labeled changes rather than estimating an angle from a stretched graph. Intercepts may lie outside the visible window.
Common errors: reversing only one subtraction, confusing x- and y-intercepts, treating A in Ax + By = C as the slope, and using a positive reciprocal rather than a negative reciprocal.
10 / UNDERSTAND & APPLY
Original practice: connect equations and graphs
All coordinates below use ordinary Cartesian axes.
ORIGINAL PRACTICE · 1
Does (3,2) lie on 2x + 5y = 16?
Reveal answer & reasoning
Yes
2(3) + 5(2) = 6 + 10 = 16.
ORIGINAL PRACTICE · 2
Find the slope of 3x + 2y = 10.
Reveal answer & reasoning
−3/2
Solving for y gives y = −(3/2)x + 5.
ORIGINAL PRACTICE · 3
Find the y-intercept of y − 4 = 2(x + 1).
Reveal answer & reasoning
(0,6)
Expand to y = 2x + 6, then set x = 0.
ORIGINAL PRACTICE · 4
Find the x-intercept of 5x − 2y = 20.
Reveal answer & reasoning
(4,0)
At an x-intercept y = 0, so 5x = 20.
ORIGINAL PRACTICE · 5
Which line through (0,−2) is parallel to y = −4x + 7?
Reveal answer & reasoning
y = −4x − 2
Parallel distinct lines share slope −4; the supplied point gives intercept −2.
ORIGINAL PRACTICE · 6
What is the slope of a line perpendicular to y = −3x + 8?
Reveal answer & reasoning
1/3
The product (−3)(1/3) is −1.
ORIGINAL PRACTICE · 7
What is the slope of y = 9?
Reveal answer & reasoning
0
All y-values are equal, so the vertical change is zero for every nonzero horizontal change.
ORIGINAL PRACTICE · 8
A tank model is L = 150 − 5t with t in minutes. When is the tank empty, and what is the valid time interval?
Reveal answer & reasoning
30 minutes; 0 ≤ t ≤ 30
Set L = 0 to obtain t = 30. The physical model starts at t = 0 and cannot represent negative remaining volume.
KEEP THE BIG IDEAS
Quick revision
- Slope is a ratio of coordinate changes with units.
- Set the other coordinate to zero to find an intercept.
- Vertical lines have undefined slope; perpendicular axis-parallel lines need separate treatment.
READ WITH CONTEXT
References & scope
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
- College Board: SAT Math content domains ↗
Official skill taxonomy reference. All teaching prose, examples, diagrams and practice in this lesson are original; this is not a past-question collection.
- College Board: SAT Math skill descriptions ↗
Factual skill-scope reference only; no question-bank questions or explanations are reproduced.
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Linear equations in two variablesSAT topic analytics ↗
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