Linear equations in two variables
These questions ask you to move between the forms of a line, write a line from given information, and read its features. Knowing which form to use for which task is most of the skill.
What College Board tests
Writing and interpreting equations of lines, often given two points, a point and a slope, or a graph. You'll also compare lines (parallel and perpendicular) and find intercepts from standard form.
Two forms to know
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (their product is ).
Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.
1 · Write a line from a point and a slope
In the xy-plane, line has a slope of 4 and passes through the point . What is the value of on line when ?
Approach The point has , so it's the y-intercept — that's . With the slope given, write and plug in .
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Answer: B
A Uses a slope of 2 (the -value) instead of 4.
C Stops at and forgets the intercept.
D Adds 3 instead of subtracting it.
2 · Write a line from two points
A line passes through and . Which equation defines the line?
Approach Two steps: first find the slope from the two points, then use one point to solve for the intercept .
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Answer: A
B Drops the negative sign on the slope.
C Right slope, but uses 10 as the intercept without solving.
D Flips the slope to its reciprocal.
3 · Parallel and perpendicular slopes
Line is defined by . Line is parallel to line . What is the slope of line ?
Approach Parallel lines never meet, which means they rise at exactly the same rate — equal slopes. Read the slope off line directly; line has the same one.
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Answer: C
A The negative reciprocal — that's the perpendicular slope, not parallel.
B Negates the slope; parallel keeps the sign.
D Flips the fraction; parallel keeps it unchanged.
4 · Find an intercept from standard form
A line is defined by . What is the x-coordinate of the point where the line crosses the x-axis?
Approach The x-axis is the line . Substitute , and the -term disappears, leaving a one-step solve for .
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Answer: C
A Divides by 6 instead of 3.
B Sets and finds the y-intercept instead.
D Reads the constant 24 without dividing.