Two-Step Equations - Pre Algebra

1. Maria solved the equation $3x - 7 = 14$ and got $x = 3$. When she substituted her answer back into the original equation to check, what value did she get for the left side of the equation?
   Answer: $2$ When substituting $x = 3$ into $3x - 7$, we get $3(3) - 7 = 9 - 7 = 2$. Since Maria's answer gives 2 on the left side but the equation should equal 14, her solution is incorrect. The correct solution is $x = 7$.
2. If $2(n - 4) = 18$, what is the value of $n - 4$?
   Answer: $9$ From $2(n - 4) = 18$, divide both sides by 2 to get $n - 4 = 9$. We don't need to solve for $n$ itself. Choice A comes from solving $n - 4 = 9$ to get $n = 13$, then incorrectly using 5. Choice C gives the value of $n$, not $n - 4$. Choice D comes from adding instead of dividing.
3. Jake's work solving $4x + 9 = 1$ is shown below: Step 1: $4x + 9 = 1$ Step 2: $4x = 1 + 9$ Step 3: $4x = 10$ Step 4: $x = \frac{10}{4} = 2.5$ In which step did Jake make his first error?
   Answer: Step 2 (incorrect operation with the constant term) In Step 2, Jake should have subtracted 9 from both sides to get $4x = 1 - 9 = -8$, not $4x = 1 + 9$. This is a common error when moving terms across the equals sign. The correct solution is $x = -2$. Steps 3 and 4 follow correctly from his error in Step 2.
4. Sarah needs to solve $7 - 2x = 21$. She decides to first add $2x$ to both sides, then subtract 21 from both sides. What equation will she have after these two steps?
   Answer: $-14 = 2x$ When solving equations, you need to carefully track what happens to both sides when you apply operations. This question tests whether you can follow a specific sequence of steps, even if they're not the most efficient path to the solution. Let's follow Sarah's steps exactly. Starting with $7 - 2x = 21$, she first adds $2x$ to both sides: - Left side: $7 - 2x + 2x = 7$ - Right side: $21 + 2x$ - Result: $7 = 21 + 2x$ Next, she subtracts 21 from both sides: - Left side: $7 - 21 = -14$ - Right side: $21 + 2x - 21 = 2x$ - Final result: $-14 = 2x$ This confirms answer choice C is correct. Looking at the wrong answers: A) $28 = 2x$ incorrectly adds 7 and 21 instead of subtracting. B) $-14 = -2x$ makes the error of keeping the negative sign on the $x$ term, forgetting that adding $2x$ to $-2x$ eliminates the negative. D) $7 = 21 + 2x$ is actually the intermediate step after just the first operation, not the final result after both steps. When working through multi-step problems, always perform operations one at a time and double-check each step before moving on. Even if someone chooses an unusual order of operations (like Sarah did here), the algebra still works as long as you apply each step correctly to both sides.
5. The perimeter of a rectangle is 36 inches. If the width is 7 inches, which equation can be used to find the length $\ell$?
   Answer: $2\ell + 14 = 36$ When you encounter rectangle perimeter problems, remember that perimeter equals the sum of all four sides. For a rectangle, this means adding length + width + length + width, which gives us the formula: Perimeter = $2\ell + 2w$. Given that the perimeter is 36 inches and the width is 7 inches, you can substitute these values into the formula: $36 = 2\ell + 2(7)$. Simplifying the width term: $36 = 2\ell + 14$. Rearranging this equation gives you $2\ell + 14 = 36$, which matches answer choice D. Let's examine why the other options are incorrect. Choice A ($2\ell + 7 = 36$) only accounts for one width instead of two, forgetting that a rectangle has two width sides. Choice B ($2(\ell + 7) = 72$) incorrectly uses 72 instead of the given perimeter of 36. Choice C ($\ell + 7 = 36$) completely ignores that perimeter involves all four sides, treating it as if you're just adding length and width once. To verify answer D is correct, you can solve it: $2\ell + 14 = 36$ means $2\ell = 22$, so $\ell = 11$. Check: $2(11) + 2(7) = 22 + 14 = 36$ ✓ Study tip: Always write out the perimeter formula first ($P = 2\ell + 2w$), then substitute the known values. This systematic approach prevents you from accidentally forgetting that rectangles have two of each side.
6. Which of the following equations is equivalent to $5 = 3y - 7$?
   Answer: $12 = 3y$ and $y = 4$ When solving linear equations, your goal is to isolate the variable by performing the same operation on both sides of the equation. This maintains the equation's balance while simplifying it step by step. Starting with $5 = 3y - 7$, you need to get $y$ by itself. First, add 7 to both sides to eliminate the $-7$ on the right side: $5 + 7 = 3y - 7 + 7$, which simplifies to $12 = 3y$. Next, divide both sides by 3 to isolate $y$: $\frac{12}{3} = \frac{3y}{3}$, giving you $y = 4$. You can verify this by substituting back: $5 = 3(4) - 7 = 12 - 7 = 5$ ✓ Choice D is correct because it shows both the intermediate step $12 = 3y$ and the final answer $y = 4$. Choice A incorrectly subtracts 7 from the left side instead of adding it to both sides. This gives $5 - 7 = -2$, not the correct $12$. Choice B shows the wrong intermediate step $-2 = 3y$ (from the error in choice A) but then correctly divides to get $y = -\frac{2}{3}$, which is wrong because the intermediate step was wrong. Choice C correctly adds 7 to get $5 + 7 = 3y$, but then makes an arithmetic error, claiming $5 + 7 = 5$ instead of $12$, leading to the incorrect answer $y = \frac{5}{3}$. Remember: always perform the same operation on both sides of an equation, and double-check your arithmetic at each step to avoid simple calculation errors.
7. For what value of $k$ does the equation $\frac{2x - k}{5} = 4$ have solution $x = 9$?
   Answer: $k = -2$ Substitute $x = 9$ into the equation: $\frac{2(9) - k}{5} = 4$, so $\frac{18 - k}{5} = 4$. Multiply by 5: $18 - k = 20$. Solve: $-k = 2$, so $k = -2$. Choice B gives $k = 2$ from sign error. Choice C comes from $18 - 7 = 11$. Choice D comes from $18 + 20 = 38$.
8. A cell phone plan charges a monthly fee of $25 plus $0.10 per text message. If someone's bill was $32.50 for one month, which equation represents the number of text messages $t$ they sent?
   Answer: $25 + 0.10t = 32.50$ When you encounter word problems about costs or pricing, your goal is to translate the real-world situation into a mathematical equation by identifying the fixed costs and variable costs. In this cell phone plan, there are two components: a fixed monthly fee of $25 (this stays the same regardless of usage) and a variable cost of $0.10 per text message (this depends on how many texts are sent). The total bill equals the fixed cost plus the variable cost, so: Total Bill = Fixed Fee + (Cost per text × Number of texts). Substituting the given values: $32.50 = 25 + 0.10t$, which can be rewritten as $25 + 0.10t = 32.50$. This matches answer choice C. Let's examine why the other options are incorrect. Choice A ($0.10t - 25 = 32.50$) subtracts the monthly fee instead of adding it, which doesn't make sense since fees increase your bill. Choice B ($25t + 0.10 = 32.50$) incorrectly multiplies the monthly fee by the number of texts and treats the per-text cost as a fixed amount. Choice D ($25(t + 0.10) = 32.50$) uses distribution incorrectly, suggesting the monthly fee is multiplied by both the number of texts and the per-text rate. Study tip: In cost problems, always identify what stays constant (fixed costs) and what changes based on usage (variable costs). Write the equation as: Total = Fixed + (Rate × Variable). This pattern appears frequently in pre-algebra word problems involving phone plans, taxi fares, and rental costs.
9. The equation $\frac{x + 5}{3} = 8$ has the same solution as which of the following equations?
   Answer: $x + 5 = 24$ Solving $\frac{x + 5}{3} = 8$: multiply both sides by 3 to get $x + 5 = 24$, then $x = 19$. Choice B gives the same equation after the first step. Choice A would give $x = 6$, Choice C would give $x = 9$, and Choice D would give $x = -\frac{7}{3}$.

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