Translating Word Problems into Equations
Half the battle on SAT word problems is turning the sentence into an equation. Read for the operation and don't drop any condition.
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
Phrases map to operations: "increased by" means add, "of" often means multiply, "is" means equals. Translate piece by piece, and make sure every condition in the sentence appears in your equation.
How to solve one, step by step
Example: "twice a number increased by 5 is 17."
- Twice a number, increased by 5: $2n + 5$.
- Is 17: $2n + 5 = 17 \Rightarrow n = 6$.
The mistakes that cost points
- Choosing the wrong operation. "Increased by" adds; "times" multiplies. Map each phrase carefully.
- Dropping a constraint. Every condition stated in the problem belongs in the equation.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
The two linear equations $y = mx - 1$ and $y - 9 + 2x = 0$ intersect at point $(2, k)$. Find the values of $m$ and $k$.
- A$m = 7$, $k = 13$
- B$m = 3$, $k = 13$
- C$m = 3$, $k = 5$
- D$m = 5$, $k = 3$
Show solution
Answer: C, $m = 3$, $k = 5$. Step 1: Rearrange the second equation to the form $y = \text{something}$. From the equation, $y = 9 - 2x$. Step 2: The point $(2, k)$ lies on this line. Substitute $x = 2$: $k = 9 - 4 = 5$. Step 3: The point $(2, 5)$ lies on the first line $y = mx - 1$. Substitute: $5 = m(2) - 1$, so $m(2) = 6$, so $m = 3$. Answer: $m = 3$, $k = 5$.
Medium
Consider the system of equations below. $3x + 2y = 20$ and $x - 2y = 4$. What is the value of $x + y$?
- A$10$
- B$9$
- C$7$
- D$6$
Show solution
Answer: C, $7$. Step 1: Add the equations term by term: $(3x + x) + (2y - 2y) = 20 + 4$, giving $4x = 24$, so $x = 6$. Step 2: Substitute into $x - 2y = 4$: $6 - 2y = 4$, so $y = 1$. Step 3: $x + y = 6 + 1 = 7$.
Hard
The two linear equations $y = mx - \frac{1}{3}$ and $y - 14 + \frac{7}{3}x = 0$ intersect at point $(3, k)$. Find the values of $m$ and $k$.
- A$m = \frac{22}{9}$, $k = 21$
- B$m = \frac{64}{9}$, $k = 21$
- C$m = 7$, $k = \frac{22}{9}$
- D$m = \frac{22}{9}$, $k = 7$
Show solution
Answer: D, $m = \frac{22}{9}$, $k = 7$. Step 1: Rearrange the second equation to the form $y = \text{something}$. From the equation, $y = 14 - \frac{7}{3}x$. Step 2: The point $(3, k)$ lies on this line. Substitute $x = 3$: $k = 14 - 7 = 7$. Step 3: The point $(3, 7)$ lies on the first line $y = mx - \frac{1}{3}$. Substitute: $7 = m(3) - \frac{1}{3}$, so $m(3) = \frac{22}{3}$, so $m = \frac{22}{9}$. Answer: $m = \frac{22}{9}$, $k = 7$.
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