Algebra Review

Rules

What you do to one side, you do the other.
Addition/Subtraction: If it is added, subtract it. If it is subtracted, add it.
Multiplication/division: If it is multiplied, divide (everything) by it. If itis divided, then multiply (everything) by it.
Exponents/Roots: If is has an exponent, take the root of both sides. If it has a root, that the exponent of both sides.
Perform addition/subtraction before multiplication/division.
Perform root/exponentials last (after multiplication/division/addition/subtraction).

Examples

Addition and Subtraction

: To solve for X, (C and Y are just numbers), we want it alone on one side. But we have a added on. By rules 1& 2, we subtract it from both sides.

$\begin{displaymath}X+C-C=Y-C\quad\rightarrow\quad X=Y-C\end{displaymath}$

$\begin{displaymath}x+2=3\quad\rightarrow\quad x+2-2=3-2\quad\rightarrow\quad x=1\end{displaymath}$

: To solve for X, we want it alone on one side. But we have a C being subtracted, so by rules 1 & 2, we add it to both sdies.

$\begin{displaymath}X-C+C=Y-C\quad\rightarrow\quad X=Y+C\end{displaymath}$

$\begin{displaymath}x-2=3\quad\rightarrow\quad x-2+2=3+2\quad\rightarrow\quad x=5\end{displaymath}$

Multiplication and Division

: To solve for X, we want it alone on one side. But we have a C multiplied to it. To solve, we divide C from both sides (Rules 1& 3).

$\begin{displaymath}\frac{XC}{C}=\frac{Y}{C}\quad\rightarrow\quad X=\frac{Y}{C}\end{displaymath}$

$\begin{displaymath}3x=12 \quad\rightarrow\quad \frac{3x}{3}=\frac{12}{3} \quad\rightarrow\quad x=4\end{displaymath}$

$\frac{X}{C}=Y$ : To solve for X, we want it alone on one side. But we have a C dividing it. To solve, we multiply both sides by C (Rules 1 & 3)

$\begin{displaymath}\frac{X\cdot C}{C}=Y\cdot C\quad\rightarrow\quad X=Y\cdot C\end{displaymath}$

$\begin{displaymath}\frac{x}{6}=12\quad\rightarrow\quad \frac{x\cdot 6}{6}=12\cdot 6 \quad\rightarrow\quad x=72\end{displaymath}$

Exponents and Roots

: To solve for X, take the $c^{th}$ root of both sides. Rules 1 & 4.

$\begin{displaymath}\sqrt[c]{X^c}=\sqrt[c]{Y}\quad\rightarrow\quad X=\sqrt[c]{Y}\end{displaymath}$

$\sqrt[c]{X}=Y$ : To solve for X, raise both sides to the power of c. Rules 1 & 4.

$\begin{displaymath}\left(\sqrt[c]{X}\right)^c=Y^c\quad\rightarrow\quad X=Y^c\end{displaymath}$

$\begin{displaymath}\sqrt[3]{X}=4\quad\rightarrow\quad X=4^3\quad\rightarrow\quad X=64\end{displaymath}$

Combinations

: To solve for X, use rules 6, 4, 2, & 1.

$\begin{displaymath}X^c+D-D=Y-D\quad\rightarrow\quad X^c=Y-D\quad\rightarrow\quad X= \sqrt[c]{Y-D}\end{displaymath}$

$\begin{displaymath}X^2+3=7\quad\rightarrow\quad X^2=4\quad\rightarrow\quad X=\sqrt{4} \quad\rightarrow\quad X=2\end{displaymath}$

$\frac{X}{C}-D=Y$ : To solve for X, use rules 1, 2, 3, & 5.

$\begin{displaymath}\frac{X}{C}-D+D=Y+D\quad\rightarrow\quad \frac{X\cdot C}{C}=C\left( Y+D\right) \quad\rightarrow\quad X=C\left( Y+D\right)\end{displaymath}$

Exponentials

If a power is raised to another power, multiply the powers.

$\begin{displaymath}\left( X^c\right)^d\quad\rightarrow\quad X^{c\dot d}\end{displaymath}$

$\begin{displaymath}\left( 10^2\right)^3\quad\rightarrow\quad 10^6\end{displaymath}$

If a root is taken of a number raised to a power, divide the power by the root.

$\begin{displaymath}\sqrt[d]{X^c}\quad\rightarrow\quad X^{\frac{c}{d}}\end{displaymath}$

$\begin{displaymath}\sqrt[3]{10^6}\quad\rightarrow\quad 10^{\frac{6}{3}}\quad\rightarrow\quad 10^2=100\end{displaymath}$

$\begin{displaymath}\sqrt[5]{X^25}\quad\rightarrow\quad X^{\frac{25}{5}}\quad\rightarrow\quad X^5\end{displaymath}$

Add powers for multiplication, subtract powers for division.

$\begin{displaymath}X^c\cdot X^d\quad\rightarrow\quad X^{c+d}\quad\quad 10^3\cdot 10^5 \quad\rightarrow\quad 10^{3+5}\quad\rightarrow\quad 10^8\end{displaymath}$

$\begin{displaymath}\frac{X^c}{X^d}\quad\rightarrow\quad X^{c-d}\quad\quad \frac{... ...{5^2} \quad\rightarrow\quad 5^{4-2}\quad\rightarrow\quad 5^2=25\end{displaymath}$