SAT Algebraic Manipulations

To successfully manipulate algebraic expressions on the SAT, you need to know how to:

• apply addition, subtraction, multiplication, and division to algebraic expressions
• apply the order of operations
• make simple substitutions

If $3m$ + $6m$ + $9m$ = $-36$, which of the following is equal to $m$?

(A) $\ \ 2$
(B) $\ \ 1$
(C) $\ \ 0$
(D) $\ \ -1$
(E) $\ \ -2$

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Correct Answer: E

Solution 1:

Tip: Follow order of operations.
Using the given equation, we solve for $m$.

$$\begin{array}{l c l l l} 3m + 6m + 9m &=& -36 &\quad \text{original expression} &(1)\\ 18m &=& -36 &\quad \text{combine like terms} &(2)\\ \frac{18m}{18} &=& \frac{-36}{18} &\quad \text{divide both sides by}\ 18 &(3)\\ m &=& -2 &\quad \text{perform division} &(4)\\ \end{array}$$

Solution 2:

Tip: Plug and check.
We plug the value of each answer choice into the given equation and select the one that doesn't yield a contradiction.

(A) If $m=2$:

$3m + 6m + 9m = 3 \cdot 2 + 6 \cdot 2 + 9 \cdot 2 = 6 +12 +18 = 36 \neq -36.$

This is a contradiction. Eliminate (A).

(B) If $m=1$:

$3m + 6m + 9m = 3 \cdot 1 + 6 \cdot 1 + 9 \cdot 1 = 3 +6 +9 = 18 \neq -36.$

This is a contradiction. Eliminate (B).

(C) If $m=0$:

$3m + 6m + 9m = 3 \cdot 0 + 6 \cdot 0 + 9 \cdot 0 = 0 \neq -36.$

This is a contradiction. Eliminate (C).

(D) If $m=-1$:

$3m + 6m + 9m = 3 \cdot (-1) + 6 \cdot (-1) + 9 \cdot (-1) = -3 -6 -9 = -18 \neq -36.$

This is a contradiction. Eliminate (D).

(E) If $m=-2$:

$3m + 6m + 9m = 3 \cdot (-2) + 6 \cdot (-2) + 9 \cdot (-2) = -6 -12 -18 = -36.$

This is correct and therefore (E) is the answer.

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Incorrect Choices:

(A)
Tip: Select the answer with the correct sign!

(B), (C), and (D)
Solution 2 explains why these choices are wrong.

If $-3(2x-5)+8 = -2x+3$, what is the value of $x$?

(A) $\ -5$

(B) $\ -\frac{5}{4}$

(C) $\ \ 0$

(D) $\ \ \frac{5}{4}$

(E) $\ \ 5$

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Correct Answser: E

Solution 1:

Tip: Follow order of operations.
We start with the given equation and we simplify.

$$\begin{array}{rcll} -3(2x-5)+8 &=& -2x+3 &\text{original equation}&\quad (1)\\ -6x+15+8 &=& -2x+3 &\text{use distributive property}&\quad (2)\\ -6x+23 &=& -2x+3 &\text{simplify}&\quad (3)\\ -6x+23-3&=&-2x+3-3 &\text{subtract} \ 3\ \text{from both sides}&\quad (4)\\ -6x+20 &=&-2x &\text{simplify}&\quad (5)\\ -6x +20+6x &=&-2x +6x&\text{add}\ 6x\ \text{to both sides}&\quad (6)\\ 20 &=& 4x &\text{combine like terms}&\quad (7)\\ \frac{20}{4} &=&\frac{4x}{4}&\text{divide both sides by} \ 4&\quad (8)\\ 5&=&x&\text{simplify the fractions}&\quad (9)\\ \end{array}$$

Solution 2:

Tip: Plug and check.
We can plug each answer choice into the given equation and check if it yields a true statement. If it does, then the choice is right. In this case, only (E) will work.

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Incorrect Choices:

(A)
Tip: Select the answer with the correct sign!
Refer to the solution above. The answer should be $5$. Selecting $-5$ would be a careless mistake.

(B)
Tip: When distributing, be careful with signs!
Refer to Solution 1 above. If in step $(2)$ we forget to distribute the negative sign, we will get:

$$\begin{array}{lcl} \fbox{-}3(2x-5)+8 &=& -2x+3 &\text{original equation}&\quad (1)\\ 6x-15+8 &=& -2x+3 &\text{mistake: didn't distribute}\\ &&&\text{negative sign}&\quad (2)\\ \end{array}$$

Simplifying, we get:

$$\begin{array}{lcl} -\frac{5}{4}&=&x&\text{simplify} \end{array}$$ But this is wrong.

(C)
Tip: The simplest choice may not be the correct one.
Plug in and check. If $x=0$, we get:

$$\begin{array}{lcl} -3(2x-5)+8 &=& -2x+3\\ -3(2\cdot0-5)+8&=&-2\cdot 0+3&\text{plug in}\ x=0\\ -3(0-5)+8&=&=0+3&\text{simplify}\\ -3(-5)+8&=&3&\text{simplify parentheses}\\ 15+8&=&3&\text{simplify the left side}\\ 23&=&3&\text{simplify the left side again}\\ \end{array}$$

But $23\neq3$. Therefore, this choice is wrong.

(D)
It is possible you made a mistake when reducing a fraction. Refer to step $(8)$ in the solution above and focus on the fraction on the left side of the equation.

$$\begin{array}{lcll} \frac{20}{4} &=&\frac{4x}{4}&\text{divide both sides by} \ 4&\quad (8)\\ \end{array}$$

We must divide both the numerator and denominator by their greatest common factor to obtain the correct reduced fraction. $4$ is the greatest number that divides both $20$ and $4$. So, $\frac{20/4}{4/4}=\frac{5}{1}=5$. But if we forget to divide the denominator by $4$, we will get this wrong answer.

If $5x+2=9$, what is the value of $5x-2$?

(A) $\ \ -9$
(B) $\ \ -5$
(C) $\ \ \frac{7}{5}$
(D) $\ \ 5$
(E) $\ \ 7$

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Correct Answer: D

Solution 1:

Tip: Look for short-cuts.
We don't need to solve for $x$ to find the answer. That's the trick. We realize that $5x-2=5x+2-4=9-4=5$.

Solution 2:

We could solve for $x$:

$$\begin{array}{rcl} 5x+2&=&9&\quad\text{given}\\ 5x&=&7&\quad\text{subtract}\ \ 2\ \text{from both sides}\\ x&=&\frac{7}{5}&\quad\text{divide both sides by}\ 5\\ \end{array}$$

Then, $5x-2=5\times\frac{7}{5}-2=7-2=5.$

So, $5x-2=5$.

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Incorrect Choices:

(A)
Tip: If you can, verify your choice.
We are given $5x+2=9$. We are looking for $5x-2$, and you may think that because the sign between $5x$ and $2$ changed, you need to change the sign of $9$ also in order to get the answer, like this: $5x-2=-9$. But verify your choice. If $5x-2=-9$, then $5x=-7$ and adding $2$ to both sides of this equation, we get $5x+2 = -7+2=-5\neq 9$. Therefore, (C) is the wrong choice.

(B)
Tip: Select the answer with the correct sign!

(C)
Tip: Read the entire question carefully.
You likely got this answer because you solved for $x$ instead of $5x-2$.

(E)
Tip: Read the entire question carefully.
You likely got this answer because you solved for $5x$, not for $5x-2$.

If you thought these examples difficult and you need to review the material, these links will help:

• Combining Like Terms
• Distributive Property
  $a(b+c)=ab+ac$
• Simple Equations
• Multi-Step Equations-Basic

• Follow order of operations.
• Select the answer with the correct sign!
• When distributing, be careful with signs!
• SAT General Tips