Educerie · SAT · Math
Algebra · ALG.2 Linear equations in two variables
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Write the equation a context describes | …which equation represents the total cost after h hours? |
| Find slope from two points | …what is the slope of the line through (2, 5) and (6, 17)? |
| Move between the three forms | Standard to slope-intercept, and back |
| Find intercepts | Set the other variable to zero |
| Recognise parallel and perpendicular lines | Same slope; slopes multiplying to −1 |
| Read a line off a graph or a table | Two points is all you need |
1The idea in one paragraph
A linear equation in two variables is a straight line, and a straight line is completely described by two numbers: where it starts and how fast it changes. Every question in this unit is one of those two numbers in disguise. Find the rate and find one point, and you can write any line the test asks for.
Slope is change in y over change in x. In a context it is always "per": lira per hour, litres per kilometre, degrees per minute.
2The three forms and what each hands you
| Form | Written | Gives you instantly |
|---|---|---|
| Slope-intercept | y = mx + b | Rate m, starting value b |
| Standard | Ax + By = C | Intercepts: put x = 0, then y = 0 |
| Point-slope | y − y₁ = m(x − x₁) | A line through a known point |
Standard form is the one students fear and it is the easy one for intercepts:
Those two points draw the line. To go from standard to slope-intercept, solve for y:
Notice the pattern: in Ax + By = C the slope is always −A/B.
3Slope from two points
Subtracting in a consistent order is the whole trick: if you take y₂ − y₁ on top you must take x₂ − x₁ underneath.
4Writing the equation from a sentence
The shape is almost always starting value + rate × variable.
The rate is negative because the quantity falls; a question whose answer choices differ only in that sign is testing exactly this.
Define the variable in writing first: t = minutes after the valve opens. Then the sentence translates itself.
5Parallel and perpendicular
Parallel: same slope, different intercept. Two parallel lines never meet, which is why a system of them has no solution.
Perpendicular: slopes multiply to −1, so one is the negative reciprocal of the other. A line perpendicular to y = (2/3)x + 1 has slope −3/2.
6Reading a table
A table of values is a line if the y values change by the same amount for equal steps in x. Take any two rows, compute the slope, then work back to x = 0 for the intercept — or use point-slope and skip that step.
Where points are lost
- Flipping the slope: dividing change in x by change in y.
- Losing the sign on a quantity that decreases.
- Reading b as the rate and m as the starting value.
- Forgetting that Ax + By = C has slope −A/B, and reporting A/B.
- Using the negative reciprocal for parallel and the same slope for perpendicular.
- Mixing up the intercepts: the y-intercept is where x = 0.
- Not defining the variable before writing a context equation.
Work it right
- Decide what is changing and what it changes with.
- Find the rate: per unit, or (y₂ − y₁)/(x₂ − x₁).
- Find one point — often the starting value at x = 0.
- Write it in whichever form the answer choices use.
- Test your equation on a second point from the question.
Try it
Q1. A line passes through (3, 8) and (7, 20). What is its equation?
A) y = 3x − 1 B) y = 3x + 1 C) y = (1/3)x + 7 D) y = 4x − 4
Q2. The equation 5x + 2y = 40 is graphed in the xy-plane. What is the y-coordinate of the y-intercept?
A) 8 B) 20 C) 40 D) −5/2
Q3. A cistern holds 900 litres and is emptied at a constant 15 litres per minute. Which equation gives the volume V, in litres, remaining after t minutes?
A) V = 15t + 900 B) V = 900 − 15t C) V = 900t − 15 D) V = 15 − 900t
Q4. Line k is perpendicular to the line y = −(4/5)x + 2. What is the slope of line k?
A) −5/4 B) −4/5 C) 4/5 D) 5/4
Q5. In the xy-plane, line m has slope 2 and passes through (−1, 4). What is the y-coordinate of the point on line m where x = 3? (Type your answer.)
In one breath
Every line is two numbers: a starting value and a rate. Get the rate from (y₂ − y₁)/(x₂ − x₁) or from the word "per", get one point from the question, and write it in whatever form the options use. Standard form Ax + By = C has slope −A/B and gives its intercepts the moment you set one variable to zero — and a quantity that decreases takes a negative rate, which is the sign the wrong answers are built on.
Answers
Q1. A — y = 3x − 1. slope first, then a point
B has the intercept's sign wrong. C inverts the slope. D comes from (20 − 8)/(7 − 4).
Q2. B — 20. set x = 0
A is the x-intercept (y = 0 gives x = 8). C is C itself. D is the slope, −5/2.
Q3. B — V = 900 − 15t. full at t = 0, falling by 15 each minute A fills rather than empties. C and D attach the rate to the wrong number entirely.
Q4. D — 5/4. negative reciprocal Flip −4/5 and change the sign: 5/4. B is the parallel slope; A flips without changing sign; C changes sign without flipping.
Q5. 12. point-slope, or count the steps
Counting works too: from x = −1 to x = 3 is four steps, each adding 2, so 4 + 8 = 12.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Algebra, skill/knowledge testing point "Linear equations in two variables"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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