Strategy: Systems, Logs, and Sequences

Systems of Equations Strategy

The ACT presents systems in word problems more often than as bare equations. The setup is the hard part.

Translation Pattern

A parking lot charges $3 per hour for the first car and$2 per hour for each additional car. A family parks 2 cars for the same number of hours and pays $40. How many hours did they park?

Setup: First car costs $3h$, second car costs $2h$, total = $40$. $3h + 2h = 40 \Rightarrow 5h = 40 \Rightarrow h = 8$

When to Use Elimination vs. Substitution

• Elimination when both equations are in standard form ($ax + by = c$) and coefficients are easy to match
• Substitution when one variable is already isolated ($y = ...$)

Logarithm Problem Strategy

Most ACT log questions test the definition or one property. Convert to exponential form:

"Solve $\log_3(x) = 4$" → $3^4 = x$ → $x = 81$

"Solve $\log_2(x) + \log_2(3) = 5$" → $\log_2(3x) = 5$ → $3x = 32$ → $x = 32/3$

Sequence Strategy

Step 1: Determine if arithmetic (constant difference) or geometric (constant ratio).

• Check: $a_3 - a_2 = a_2 - a_1$? → Arithmetic.
• Check: $a_3 / a_2 = a_2 / a_1$? → Geometric.

Step 2: Find $d$ or $r$, then use the appropriate formula.

ACT Shortcut: For "find the sum of the first $n$ terms" of an arithmetic sequence, use $S_n = \frac{n}{2}(\text{first term} + \text{last term})$. You don't need every term — just the first and last.

Backsolving Strategy

Since the ACT is all multiple choice, you can often plug answer choices back into the problem to see which one works. This is especially useful for:

• Systems where setup is confusing
• Function equations where algebraic manipulation is complex
• Any question where you're stuck and running low on time

Start with choice B or C (the middle values) to determine which direction to go.