# Page 3 Elementary Algebra Study Guide for the ACCUPLACER® test

### Operations with Algebraic Expressions

Topics related to the second type of ACCUPLACER test item are:

#### Evaluating Simple Formulas and Expressions

Simple formulas and expressions will often involve a single variable and a few mathematical operations:

This is an algebraic expression involving a variable, x, that when multiplied by 3 and then reduced by 2 is equal to the number 10. Algebraic expressions in one variable can be solved by combining like terms such that the variable is isolated and set equal to a number. Solving the equation above:

$3x - 2 = 10$
$3x - 2 + 2 = 10 + 2$
$3x = 12$
$\frac{3x}{3} = \frac{12}{3} -> x = 4$

This can be verified by plugging the value of 4 back into the original expression and confirming that the result on the left is equal to the result on the right. And indeed, $3(4) - 2 = 10 -> 12 - 2 = 10 -> 10 = 10$. Verifying the validity of a solved variable is an invaluable strategy for confirming solutions on exams.

The general approach for solving an algebraic expression for a variable is to undo all of the operations that are being performed on the variable, starting from the last operation and working back to the first operation:

The last operation performed to the variable x is the multiplication by 2 outside of the parenthesis. So, this is the first operation that must be undone to isolate the variable. Divide both sides by 2 to undo the multiplication by 2. If this operation is not performed on both sides of the equal sign, the integrity of the original expression will be lost- remember that in order to keep the original equality true, any operation performed on the left hand side must also be performed on the right hand side.

Notice that the parenthesis can be removed after dividing both sides by 2 because there is no longer an operation being performed on the expression as a whole. The last operation being performed on the variable is now the subtraction by 2. To undo the subtraction by 2, add 2 to both sides of the equation.

An early encounter with isolating a variable with a fractional coefficient often involves the 2 step process of multiplying and then dividing both sides of the equation. However, a familiarity with the reciprocal of a fraction enables the isolation of the variable in a single step. The reciprocal of a fraction is another fraction that when multiplied by the first is equal to 1:

To isolate the variable above, multiply both sides of the equation by the reciprocal of $\frac{3}{5}$

Not all algebraic expressions involve a single variable. In such cases, you may be asked to solve for one of the variables included:

Solve for r:
$V = \frac{4}{3} \cdot \pi \cdot r^3$

As before, solve for the variable by undoing the operations performed on it in. In this case, the variable r is first raised to the third power, then multiplied by $\pi$, and then multiplied by $\frac{4}{3}$. Isolate r by first multiplying both sides of the equation by the reciprocal of the fraction:

$\frac{3}{4} \cdot V = \frac{3}{4} \cdot \frac{4}{3} \cdot \pi \cdot r^3$ $\frac{3}{4} \cdot V = \pi \dot r^3$

Divide both sides by $\pi$ $\frac{1}{\pi} \cdot \frac{3}{4} \cdot V = r^3$

To isolate a single r, eliminate the 3rd power by taking the cube root of both sides:

$\sqrt[3]{\frac{1}{\pi} \cdot \frac{3}{4} \cdot V} = r$.

Expressions involving more than one variable can only be solved in terms of the other variables in the expression.