Worked Examples: SAT Algebra

Example 1: Rate Problem (Medium Difficulty)

A pool is being filled with two hoses. Hose A fills at 3 gallons per minute, and Hose B fills at 5 gallons per minute. The pool already contains 40 gallons. Which equation gives the total gallons $G$ in the pool after $m$ minutes with both hoses running?

Solution:

• Starting amount: 40 gallons
• Rate with both hoses: $3 + 5 = 8$ gallons/minute
• After $m$ minutes: $G = 40 + 8m$

The slope is 8 (combined fill rate), the intercept is 40 (initial water).

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Example 2: System Setup (Hard Difficulty)

A theater sells adult tickets for $12 and child tickets for$8. For a show, 350 tickets were sold for a total of $3,400. How many adult tickets were sold?

Setup: Let $a$ = adult tickets, $c$ = child tickets. $a + c = 350$ $12a + 8c = 3{,}400$

Solve by elimination: Multiply first equation by $-8$: $-8a - 8c = -2{,}800$ $12a + 8c = 3{,}400$

Add: $4a = 600 \Rightarrow a = 150$.

Answer: 150 adult tickets.

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Example 3: Slope Interpretation (Easy–Medium)

The equation $d = 65t + 20$ models the distance $d$ in miles from a city after $t$ hours of driving. What does the 20 represent?

The 20 is the y-intercept ($t = 0$). It represents the initial distance from the city when driving begins — the driver started 20 miles away.

Common trap: A distractor will say "the speed of the car" (that's the 65, the slope).

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Example 4: Inequality (Medium)

A student needs at least 480 points across 6 tests. After 5 tests, the student has 390 points. Which inequality represents the score $s$ needed on the 6th test?

$390 + s \geq 480$ $s \geq 90$

Key: "at least" → $\geq$, not $>$.

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Example 5: No Solution System (Hard)

For what value of $a$ does the system below have no solution? $ax - 6y = 10$ $3x - 2y = 5$

For no solution, the ratios of coefficients must be equal but not equal to the ratio of constants: $\frac{a}{3} = \frac{-6}{-2} \neq \frac{10}{5}$ $\frac{a}{3} = 3 \Rightarrow a = 9$

Check: $\frac{10}{5} = 2 \neq 3$. Confirmed — when $a = 9$, the lines are parallel.