Solving Equations

$\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} $

Section 1.4 Solving Equations

Fact 1.4.0.14. Order of Operations When Solving.

When we are trying to solve an equation, we are working backwards to uncover what the variable must have been. Since we are working backwards, we use the order of operations backwards.

🔗

Example 1.4.0.15.

Suppose we want to solve the following equation for $x\text{:}$

🔗

\begin{equation*} 3(x+7)^3-2=22 \end{equation*}

We will apply the same thing to both sides, each time deciding what to cancel based on the order of operations. For example, our normal GEMDAS would have Add/Subtract last, which means it’s the first thing we need to deal with. We have a $-2$ at the end of this equation, so since adding cancels subtraction, we will add 2 to both sides. We continue each step just like this:

🔗

\begin{align*} 3(x+7)^3-2\amp=22\\ 3(x+7)^3-2{\color{red}{+2}}\amp=22{\color{red}{+2}}\\ 3(x+7)^3\amp=24\\ \frac{3(x+7)^3}{\color{red}{3}}\amp=\frac{24}{\color{red}{3}}\\ (x+7)^3\amp=8\\ \sqrt[3]{(x+7)^3}\amp=\sqrt[3]{8}\\ x+7\amp=2\\ x+7{\color{red}{-7}}\amp=2{\color{red}{-7}}\\ x\amp=-5 \end{align*}

🔗

Prev Top Next