Parallel and Perpendicular Lines
Two lines' slopes tell you instantly whether they're parallel or perpendicular — if you remember the right rule.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals — flip the fraction and change the sign.
How to solve one, step by step
Example: a line has slope $\frac{2}{3}$.
- A parallel line also has slope $\frac{2}{3}$.
- A perpendicular line has slope $-\frac{3}{2}$ (flip and negate).
The mistakes that cost points
- Just negating, not reciprocating. Perpendicular slopes are negative reciprocals — flip the fraction too.
- Using the same slope for perpendicular lines. Equal slopes mean parallel, not perpendicular.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A school store sells notebooks for $\$2$ each and pens for $\$1$ each. Maya buys $k$ notebooks and $3$ pens and spends exactly $\$11$. Which equation correctly models this situation?
- A$2k + 1 = 11$
- B$2k + 3 = 11$
- C$2k \times 3 = 11$
- D$2 + 3k = 11$
Show solution
Answer: B, $2k + 3 = 11$. Cost of notebooks: $2k$. Cost of $3$ pens: $3 \times 1 = 3$. Total: $2k + 3 = 11$. Solving: $2k = 8 \Rightarrow k = 4$ notebooks.
Medium
Two friends are saving money. Arturo starts with $\$120$ and saves $\$20$ per week. Brianna starts with $\$30$ and saves $k$ dollars per week. After $6$ weeks they have saved the same total amount. Which equation correctly models this situation?
- A$120 + 20(6) = 30 + 6k$
- B$20(6) = 30 + 6k$
- C$120 + 20 = 30 + k$
- D$120(20) = 30(k)$
Show solution
Answer: A, $120 + 20(6) = 30 + 6k$. Arturo's total after $6$ weeks: $120 + 20(6) = 120 + 120 = 240$. Set equal to Brianna's total: $30 + 6k = 240 \Rightarrow 6k = 210 \Rightarrow k = 35$.
Hard
A contractor charges a fixed consultation fee of $\$c$ plus $\$75$ per hour of work. A competing contractor charges no consultation fee but $\$100$ per hour. For a job requiring $k$ hours, both contractors charge the same total amount. If the first contractor's consultation fee is $\$150$, which equation correctly models the number of hours at which costs are equal?
- A$150 + 75k = 100k$
- B$75k = 100k + 150$
- C$150 + 75 = 100k$
- D$150k + 75 = 100k$
Show solution
Answer: A, $150 + 75k = 100k$. First contractor's total: $150 + 75k$. Second contractor's total: $100k$. Set equal: $150 + 75k = 100k \Rightarrow 150 = 25k \Rightarrow k = 6$ hours.
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