# 9.2: Definition of an Equation

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\section{Definition of an Equation}%section 1

An {\bf \large equa}tion is a mathematical statement that has two expressions. They are separated by an {\bf \large equa}l sign. The to sides have the same value. The two sides of the equation are called members.

A linear equation in one variable uses expoent $1$ only on the variable.

$5+3=8$

$x+3=8$

are examples of equations.

Think of an equation as a balanced scale.

If a balance is in equilibrium, adding and/or subtracting the same number to/from both scales will keep the balance in equilibrium.

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\section{Reversing the Operation of Addition/Subtraction}%section 2

Being able to add or subtract the same number to or from both sides of an equation is straightforward. An addition can be used to cancel (void, reverse) the effect of a subtraction.

If $x+4=9$

then

$x+4-4=9-4$

or

$x=5$.

In this example one can guess the solution $x=5$ by observation. A solution of an equation (usually) is a number that satisfies the equation, that is, makes the two sides equal.

You may have been taught to take this thing (like a term) from one side (a member) and throw it on the other side with the sign changed. One year (or one month, or one week) from now you may question whether you need to change the sign.

Adding/subtracting the same expression from both sides of a balance in equilibrium can be remembered for a longer period of time because it makes sense. It is more logical than the voodoo pingpong mathematics.

The simplification of equations by addition/subtraction is extremely useful in more complicated situations. Be careful later when you need to simplify (void, reverse) multiplication or division. These latter operations cannot be undone by either addition or subtraction.

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Example 1:

Is $-3$ a solution of $2x+6=12$?

{\bf Solution:}

$\begin{array}{rcl lll} 2x+6&=&12\\[5pt] 2()+6&=&12&\hbox{Ready to receive the value for the unknown.}\\[5pt] 2(-3)+6&=&12&\hbox{Introduce the value.}\\[5pt] -6+6&=&12&\hbox{Multiply.}\\[5pt] 0&=&12&\hbox{The left side is not equal to the right side. }\\[5pt] &&&\hbox{$-3$is not a solution of the equation.}\\ \end{array}$

\newpage

Example 2:\\[5pt]

Is $3$ a solution of $2y+6=12$?\\[5pt]

{\bf Solution:}\\[5pt]

$\begin{array}{rcl lll} 2y+6&=&12\\[4pt] 2()+6&=&12&\hbox{Ready to receive the value for the unknown.}\\[4pt] 2(3)+6&=&12&\hbox{Introduce the value.}\\[4pt] 6+6&=&12&\hbox{Multiply.}\\[4pt] 12&=&12&\hbox{Left side$=$right side.$3$is a solution} \end{array}$

Example 3:\\[5pt]

Is $1.3$ a solution of $3(x-0.8)+2.5=2.7+x$?\\[5pt]

{\bf Solution:}\\[5pt]

$\begin{array}{rcl lll} 3(x-0.8)+2.5&=&2.7+x\\[4pt] 3(()-0.8)+2.5&=&2.7+()&\hbox{Ready to get the value for the unknown.}\\[4pt] 3((1.3)-0.8)+2.5&=&2.7+(1.3)&\hbox{Introduce the value.}\\[4pt] 3(1.3-0.8)+2.5&=&2.7+1.3&\hbox{Subtract}.\\[4pt] 3(0.5)+2.5&=&4.0&\hbox{Multiply.}\\[4pt] 1.5+2.5&=&4.0&\hbox{Add.}\\[4pt] 4.0&=&4.0&\hbox{Left side$=$right side.$1.3$is a solution} \end{array}$

It is a good idea to end up with all the unknowns on one side and the constants on the other side.

Example 4:

Is $\displaystyle \frac{1}{8}$ a solution of $\displaystyle 3x+5x-4(1-x)=\displaystyle -20x$?

{\bf Solution:}

$\begin{array}{rcl lll} \displaystyle 3x+5x-4(1-x)&=&\displaystyle -20x\\[5pt] \displaystyle 3\left(\right)+5()-4(1-())&=&\displaystyle -20()\ \ \ \hbox{Ready to get the value for the unknown.}\\[8pt] \displaystyle 3\left(\displaystyle \frac{1}{8}\right)+5\left(\displaystyle \frac{1}{8}\right)-4\left(1-\left(\displaystyle \frac{1}{8}\right)\right)&=&\displaystyle -20\left(\displaystyle \frac{1}{8}\right)\ \ \ \hbox{Populate.}\\[10pt] \displaystyle \frac{3}{8}+\displaystyle \frac{5}{8}-4\left(\displaystyle \frac{8}{8}-\displaystyle \frac{1}{8}\right)&=&\displaystyle \frac{-5}{2}\ \ \ \hbox{Prepare fraction subtraction, reduce right side}\\[10pt] \displaystyle \frac{3+5}{8}-4\left(\displaystyle \frac{7}{8}\right)&=&\displaystyle \frac{-5}{2}\ \ \ \hbox{Subtraction.}\\[10pt] \displaystyle \frac{8}{8}-\displaystyle \frac{7}{2}&=&\displaystyle \frac{-5}{2}\ \ \ \hbox{Reduce.}\\[10pt] \displaystyle \frac{2}{2}-\displaystyle \frac{7}{2}&=&\displaystyle \frac{-5}{2}\ \ \ \hbox{Reduce.}\\[10pt] -\displaystyle \frac{5}{2}&=&\displaystyle \frac{-5}{2}\ \ \ \hbox{Reduce. Left side is = to right side.}\\[10pt] \end{array}$\\
$\displaystyle \frac{1}{8}$ is a solution of the equation.\\[10pt]

Example 5:

Solve $t-6=13$.

{\bf Solution:}

$\begin{array}{rcl lll} t-6&=&13\\[5pt] t-6+6&=&13+6&\hbox{added to both sides.}\\[5pt] t&=&19 \end{array}$

Example 6:

Solve $3x-\displaystyle \frac{2}{9}=2x+\displaystyle \frac{5}{12}$.

{\bf Solution:}

$\begin{array}{rcl lll} 3x-\displaystyle \frac{2}{9}&=&\ \ 2x+\displaystyle \frac{5}{12}\\[15pt] -2x\ \ \ \ \ \ &&-2x&\hbox{Subtract$2x$from both sides.}\\$ $\\[-8pt] \cline{1-3}$ $\\[-8pt] x-\displaystyle \frac{2}{9}&=&\displaystyle \frac{5}{12}\\[15pt] \displaystyle \frac{2}{9}&&\displaystyle \frac{2}{9}&\hbox{Add$\displaystyle \frac{2}{9}$to both sides.}\\$ $\\[-8pt] \cline{1-3}$ $\\[-8pt] x&=&\displaystyle \frac{5}{12}+\frac{2}{9}\\[15pt] x&=&\displaystyle \frac{5}{12}+\frac{2}{9}&\hbox{The LCD is$36$}\\[15pt] x&=&\displaystyle \frac{5\cdot 3}{12\cdot 3}+\frac{2\cdot 4}{9\cdot 4}\\[15pt] x&=&\displaystyle \frac{15}{36}+\frac{8}{36}\\[15pt] x&=&\displaystyle \frac{23}{36} \end{array}$

Multiplying both sides of the original equation by the LCD is usually a better way of solving an equation with fractions. We'll come back to this problem.

Example 7:

Solve $5x-9+4x-2=8x+11-22$.

{\bf Solution:}

$\begin{array}{rcl lll} 5x-9+4x-2&=&8x+11-22\\[5pt] (5x+4x)-(9+2)&=&8x-(22-11)&\hbox{Combine like-terms}\\[5pt] 9x-11&=&8x-11&\hbox{Compute.}\\[5pt] 11&&\ \ \ \ \ \ 11&\hbox{Add 11 to both sides.}\\$ $\\[-8pt] \cline{1-3}$ $\\[-8pt] 9x&=&\ \ 8x&\hbox{Compute.}\\ -8x&&-8x&\hbox{Subtract$8x$from both sides.}\\$ $\\[-8pt] \cline{1-3}$ $\\[-8pt] x&=&0 \end{array}$

Example 8:

President Hayes died at a certain age. President Adams died at age $20$ years more than Hayes. President Theodore Roosevelt died at an age $10$ years less than Hayes. The sum of all three ages equals $90$ more than the sum of Hayes' and Roosevelt's ages.

How old was President Hayes when he died?

Hint: Let $x$ be the age at which President Hayes died.

{\bf Solution:}

President Adams' age at death: $x+20$\\[5pt]
President Hayes' age at death: $x$\\[5pt]
President Roosevelt's age at death: $x-10$\\[5pt]
The sum of all three ages: $x+20+x+x-10=3x+10$.\\[5pt]
90 more than the sum of Hayes and Roosevelt: $x+x-10+90=2x+80$.\\[5pt]
Equation:\\[5pt]
$\begin{array}{rcl lll} 3x+10&=&2x+80&\hbox{All 3 ages$=90$more than the sum of Hayes' and Roosevelt's ages}\\[5pt] -10&&\ \ \ \ -10\\$ $\\[-8pt] \cline{1-3}$ $\\[-8pt] 3x&=&\ \ 2x+70\\[5pt] -2x&&-2x\\$ $\\[-8pt]\cline{1-3}$ $\\[-8pt] x&=&70 \end{array}$

President Hayes was $70$ years old when he died.

\section{Exercises 9}%section

\begin{enumerate}
\item %%% 1
Is $-5$ a solution of $6x-8=38$?\\

\item %%% 2
Is $1$ a solution of $2y-14=12(y-2)$?\\

\item %%% 3
Is $2.1$ a solution of $5(x-0.6)-5.2=2.9-x$?\\

\item %%% 4
Is $\displaystyle \frac{1}{6}$ a solution of $\displaystyle 2x+4x-6(1-x)=\displaystyle -24x$?\\

\item %%% 5
Solve $p-7=22$.\\

\item %%% 6
Solve $4y-\displaystyle \frac{3}{10}=3y+\displaystyle \frac{4}{15}$.\\

\item %%% 7
Solve $5(x-9)+4(x-2)=8(x+11)$.

\item %%% 8
The gestation for an elk is a certain number. \\[5pt]
The gestation for a chimpanzee is $20$ days less.\\[5pt]
The gestation for a horse is $80$ days more than an elk.\\[5pt]
The sum of all three gestations equals $150$ less than twice the sum of the gestations of a chimpanzee and an elk. \\[5pt]
What is the gestation of an elk?\\[5pt]
Hint: Let $x$ be the gestation of an elk.\\[10pt]
\end{enumerate}

\fbox{\Huge STOP!}

\begin{enumerate}
\item %%% 1
Is $-5$ a solution of $6x-8=38$?\\[5pt]
{\bf Solution:}\\[5pt]
$\begin{array}{rcl lll} 6x-8&=&38\\[5pt] 6()-8&=&38&\hbox{Prepare to receive a value.}\\[5pt] 6(-5)-8&=&38&\hbox{Populate and multiply}\\[5pt] -30-8&=&38&\hbox{Subtract.}\\[5pt] -38&=&38&\hbox{The right side is not equal to the left side. }\\[5pt] \end{array}$\\[4pt]
$x=-5$ is not a solution.

\item %%% 2
Is $1$ a solution of $2y-14=12(y-2)$?\\[5pt]
{\bf Solution:}\\[5pt]
$\begin{array}{rcl lll} 2y-14&=&12(y-2)\\[5pt] 2()-14&=&12(()-2)&\hbox{Prepare to receive a value.}\\[5pt] 2(1)-14&=&12((1)-2)&\hbox{Populate.}\\[5pt] 2-14&=&12(-1)&\hbox{Compute.}\\[5pt] -12&=&-12&\hbox{The right side is equal to the left side.}\\[5pt] \end{array}$\\[5pt]
$y=1$ is a solution of the equation.

\item %%% 3
Is $2.1$ a solution of $5(x-0.6)-5.2=2.9-x$?

{\bf Solution:}

$\begin{array}{rcl lll} 5(x-0.6)-5.2&=&2.9-x\\[5pt] 5(()-0.6)-5.2&=&2.9-()&\hbox{Prepare to receive a value.}\\[5pt] 5((2.1)-0.6)-5.2&=&2.9-(2.1)&\hbox{Populate.}\\[5pt] 5(2.1-0.6)-5.2&=&2.9-2.1&\hbox{Compute.}\\[5pt] 5(1.5)-5.2&=&0.8&\hbox{Compute.}\\[5pt] 7.5-5.2&=&0.8&\hbox{Compute.}\\[5pt] 2.3&=&0.8&\hbox{The right side is not equal to the left side. } \end{array}$\\[8pt]
$x=2.1$ is not solution of the equation.

\item %%% 4
Is $\displaystyle \frac{1}{6}$ a solution of $\displaystyle 2x+4x-6(1-x)=\displaystyle -24x$?\\[15pt]
{\bf Solution:}\\[13pt]
$\begin{array}{rcl lll} \displaystyle 2x+4x-6(1-x)&=&\displaystyle -24x\\[13pt] \displaystyle 2()+4()-6(1-())&=&-24\displaystyle ()\\[20pt] \displaystyle 2\left(\displaystyle \frac{1}{6}\right)+4\left(\displaystyle \frac{1}{6}\right)-6\left(1-\left(\displaystyle \frac{1}{6}\right)\right)&=&24\left(\displaystyle \frac{1}{6}\right)\\[20pt] \displaystyle \frac{2}{6}+\displaystyle \frac{4}{6}-6\left(\displaystyle \frac{6}{6}-\displaystyle \frac{1}{6}\right)&=&-4\\[20pt] \displaystyle \frac{2+4}{6}-\displaystyle \frac{6(6-1)}{6}&=&-4\\[20pt] \displaystyle 1-\displaystyle 5&=&-4\\[13pt] \displaystyle -4&=&-4&\hbox{The left side equals the right side. } \end{array}$\\
Yes, $\displaystyle \frac{1}{6}$ is a solution.

\item %%% 5
Solve $p-7=22$.\\[10pt]
{\bf Solution:}\\[10pt]
$\begin{array}{rcl lll} p-7&=&22\\ 7&&\ \ 7\\$ $\\[-5pt] \cline{1-3}$ $\\[-5pt] p&=&29 \end{array}$\\[10pt]

\item %%% 6
Solve $4y-\displaystyle \frac{3}{10}=3y+\displaystyle \frac{4}{15}$.\\[10pt]
{\bf Solution:}\\[10pt]
$\begin{array}{rcl lll} 4y-\displaystyle \frac{3}{10}&=&\ \ 3y+\displaystyle \frac{4}{15}\\[15pt] -3y\ \ \ \ \ \ \ &&-3y\\$ $\\[-1pt] \cline{1-3}$ $\\[-1pt] y-\displaystyle \frac{3}{10}&=&\displaystyle \frac{4}{15}\\[15pt] \displaystyle \frac{3}{10}&&\displaystyle \frac{3}{10}\\$ $\\[-8pt] \cline{1-3}$ $\\[-1pt] y&=&\displaystyle \frac{4}{15}+\displaystyle \frac{3}{10}\\[15pt] y&=&\displaystyle \frac{4\cdot 2}{15\cdot 2}+\displaystyle \frac{3\cdot 3}{10\cdot 3}\\[15pt] y&=&\displaystyle \frac{8}{30}+\displaystyle \frac{9}{30}\\[15pt] y&=&\displaystyle \frac{8+9}{30}\\[15pt] y&=&\displaystyle \frac{17}{30} \end{array}$\\[20pt]

\item %%% 7
Solve $5(x-9)+4(x-2)=8(x+11)$.\\[10pt]
{\bf Solution:}\\[10pt]
$\begin{array}{rcl lll} 5(x-9)+4(x-2)&=&8(x+11)\\[5pt] 5x-5(9)+4x-4(2)&=&8x+8(11)\\[5pt] 5x-45+4x-8&=&8x+88\\[5pt] (5x+4x)-(45+8)&=&8x+88\\[5pt] 9x-53&=&\ \ 8x+88\\\\[-5pt] -8x\ \ \ \ \ &&-8x\\$ $\\[-5pt] \cline{1-3}$ $x-53&=&88\\[5pt] +53&&+53\\$ $\\[-5pt] \cline{1-3}$ $x&=&88+53\\[5pt] x&=&141 \end{array}$\\

\newpage

\item %%% 8
The gestation for an elk is a certain number. \\[5pt]
The gestation for a chimpanzee is $20$ days less.\\[5pt]
The gestation for a horse is $80$ days more than an elk.\\[5pt]
The sum of all three gestations equals $150$ less than twice the sum of the gestations of a chimpanzee and an elk. \\[5pt]
What is the gestation of an elk?\\[5pt]
Hint: Let $x$ be the gestation of an elk.\\[10pt]
{\bf Solution:}\\[10pt]
Gestation time of an elk: $x$.\\[5pt]
Gestation time of a chimpanzee: $x-20$.\\[5pt]
Gestation time of a horse: $x+80$.\\[5pt]
Sum of all three gestations: $x+x-20+x+80=3x+60$\\[5pt]
Sum of gestations of chimpanzee \& elk: $\! x\!-20\!+\!x\!=\!2x\!-\!20$.\\[5pt]
Twice the last sum: $2(2x-20)=4x-40$.

Equation:

$\begin{array}{rcl lll} 3x+60&=& 4x-40-150&\hbox{Sum of$3$gest.$=150$less than twice gest. of (chimp$+$elk).}\\[5pt] 3x+60&=& 4x-190\\[5pt] \!\!\!\!-3x\ \ \ \ \ \ &&\!\!-3x\\$ $\\[-8pt] \cline{1-3}$ $\\[-8pt] 60&=&x-190\\[5pt] 190&&\ \ \ \ \ 190\\$ $\\[-8pt]\cline{1-3}$ $\\[-8pt] 250&=&x\\[5pt] \end{array}$\\[10pt]
The gestation of an elk is $250$ days.

\end{enumerate}

%\end{document}

This page titled 9.2: Definition of an Equation is shared under a not declared license and was authored, remixed, and/or curated by Henri Feiner.