Multi-Step Equations - SSAT Middle Level Quantitative

1. What is the solution to $7 - 2(x + 1) = 3x + 8$?
   Answer: $x = -\frac{3}{5}$ This is a linear equation problem that tests your ability to distribute, combine like terms, and isolate the variable. When you see an equation with parentheses and variables on both sides, work systematically through the order of operations. Start by distributing the $-2$ on the left side: $7 - 2(x + 1) = 7 - 2x - 2 = 5 - 2x$. Now your equation becomes $5 - 2x = 3x + 8$. Next, collect all terms with $x$ on one side by adding $2x$ to both sides: $5 = 5x + 8$. Then subtract $8$ from both sides: $-3 = 5x$. Finally, divide by $5$: $x = -\frac{3}{5}$. You can verify this by substituting back into the original equation. Choice A ($x = -\frac{3}{5}$) is correct. Choice B ($x = \frac{3}{5}$) likely results from a sign error when moving terms across the equal sign—a very common mistake. Choice C ($x = -1$) might come from incorrectly distributing or combining like terms, perhaps treating the equation as $7 - 2x - 1 = 3x + 8$ instead of properly distributing. Choice D ($x = 1$) could result from multiple sign errors that coincidentally produce a clean integer answer. Always double-check your distribution step and be extra careful with signs when moving terms from one side to the other. The SSAT often includes answer choices that reflect common algebraic mistakes, so substituting your answer back into the original equation is a reliable way to catch errors.
2. If $4(2y - 3) + y = 3(y + 1) + 11$, what is $y$?
   Answer: $y = \frac{13}{3}$ Distribute on both sides: $4(2y - 3) + y$ becomes $8y - 12 + y = 9y - 12$, and $3(y + 1) + 11$ becomes $3y + 3 + 11 = 3y + 14$. So $9y - 12 = 3y + 14$. Subtract $3y$ from both sides: $6y - 12 = 14$. Add 12: $6y = 26$. Divide by 6: $y = \frac{26}{6} = \frac{13}{3}$. Check: the left side is $4\left(\frac{26}{3} - 3\right) + \frac{13}{3} = \frac{68}{3} + \frac{13}{3} = 27$, and the right side is $3\left(\frac{16}{3}\right) + 11 = 27$. The answer $y = 4$ comes from multiplying only the $y$ by 3 and not the 1, which gives $3y + 12$ on the right and $6y = 24$. Substituting $y = 4$ into the real equation gives 24 on the left and 26 on the right. The other whole-number answers fail the same check. Not every equation has a whole-number solution, so trust your algebra and verify by substituting.
3. If $5(g - 3) + 2g = g + 9$, then $g$ equals
   Answer: $g = 4$ This is a linear equation that requires you to isolate the variable $g$ by using algebraic manipulation techniques. Start by expanding the left side of the equation. Distribute the 5: $5(g - 3) = 5g - 15$. Now your equation becomes: $5g - 15 + 2g = g + 9$. Combine like terms on the left side: $5g + 2g = 7g$, so you have $7g - 15 = g + 9$. To isolate $g$, subtract $g$ from both sides: $7g - g - 15 = g - g + 9$, which simplifies to $6g - 15 = 9$. Add 15 to both sides: $6g = 24$. Finally, divide both sides by 6: $g = 4$. Let's verify: Substitute $g = 4$ back into the original equation. Left side: $5(4 - 3) + 2(4) = 5(1) + 8 = 13$. Right side: $4 + 9 = 13$. ✓ Looking at the wrong answers: Choice A ($g = 2$) would give you $5(-1) + 4 = -1$ on the left and $11$ on the right—not equal. Choice C ($g = 6$) produces $5(3) + 12 = 27$ versus $15$—too large. Choice D ($g = 8$) gives $5(5) + 16 = 41$ versus $17$—way too large. The answer is B. Strategy tip: Always check your answer by substituting back into the original equation. This catches arithmetic errors and confirms you've solved correctly—a crucial habit for avoiding careless mistakes on the SSAT.

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