<< Chapter < Page | Chapter >> Page > |
Solve for a: $-4a+2-a=3+5a-2$ $$\begin{array}{cccc}{-}{4}{a}+2{-}{a}& =& {3}+5a{-}{2}& \mathit{\text{Add same side like terms first.}}\\ -5a+2& =& 5a+1& \hfill \\ -5a+2{-}{5}{a}& =& 5a+1{-}{5}{a}& \mathit{\text{Subtract5afrombothsides}}\text{.}\hfill \\ -10a+2& =& 1& \\ -10a+2{-}{2}& =& 1{-}{2}& \mathit{\text{Subtract2frombothsides}}\text{.}\hfill \\ -10a& =& -1& \\ \frac{-10a}{{-}{10}}& =& \frac{-1}{{-}{10}}& \mathit{\text{Dividebothsidesby-10}}\text{.}\hfill \\ a& =& \frac{1}{10}& \end{array}$$ The solution set is $\left\{\frac{1}{10}\right\}$
When solving linear equations the goal is to determine what value, if any, will solve the equation. A general guideline is to use the order of operations to simplify the expressions on both sides first.
Solve for x: $5(3x+2)-2=-2(1-7x)$ $$\begin{array}{cccc}5(3x+2)-2& =& -2(1-7x)& \mathit{\text{Distribute}}\text{.}\hfill \\ 15x{+}{10}{-}{2}& =& -2+14x& \mathit{\text{Addsamesideliketerms.}}\hfill \\ 15x+8& =& -2+14x& \\ 15x+8{-}{14}{x}& =& -2+14x{-}{14}{x}& \mathit{\text{Subtract14xonbothsides.}}\hfill \\ x+8& =& -2& \\ x+8{-}{8}& =& -2{-}{8}& \mathit{\text{Subtract8onbothsides.}}\hfill \\ x& =& -10& \end{array}$$ The solution set is $\left\{-10\right\}$ .
There are three different kinds of equations defined as follows.
So far we have seen only conditional linear equations which had one value in the solution set. If when solving an equation and the end result is an identity, like say 0 = 0, then any value will solve the equation. If when solving an equation the end result is a contradiction, like say 0 = 1, then there is no solution.
Solve for x: $4(x+5)+6=2(2x+3)$ $$\begin{array}{cccc}4(x+5)+6& =& 2(2x+3)& \mathit{\text{Distribute}}\hfill \\ 4x{+}{20}{+}{6}& =& 4x+6& \mathit{\text{Add same side like terms}}\text{.}\hfill \\ 4x+26& =& 4x+6& \\ 4x+26{-}{4}{x}& =& 4x+6{-}{4}{x}& \mathit{\text{Subtract 4x on both sides.}}\hfill \\ 26& =& 6& \mathit{\text{False}}\hfill \end{array}$$ There is no solution, $\varnothing $ .
Solve for y: $3(3y+5)+5=10(y+2)-y$ $$\begin{array}{cccc}3(3y+5)+5& =& 10(y+2)-y& \mathit{\text{Distribute}}\hfill \\ 9y{+}{15}{+}{5}& =& {10}{y}+20{-}{y}& \mathit{\text{Addsamesideliketerms}}\text{.}\hfill \\ 9y+20& =& 9y+20& \\ 9y+20{-}{20}& =& 9y+20{-}{20}& \mathit{\text{Subtract20onbothsides}}\text{.}\hfill \\ 9y& =& 9y& \\ 9y{-}{9}{y}& =& 9y{-}{9}{y}& \mathit{\text{Subtract9yonbothsides}}\text{.}\hfill \\ 0& =& 0& \mathit{\text{True}}\hfill \end{array}$$ The equation is an identity, the solution set consists of all real numbers, $\Re $ .
Literal equations, or formulas, usually have more than one variable. Since the letters are placeholders for values, the steps for solving them are the same. Use the properties of equality to isolate the indicated variable.
Solve for a: $P=2a+b$ $$\begin{array}{cccc}P& =& 2a+b& \\ P{-}{b}& =& 2a+b{-}{b}& \mathit{\text{Subtractbonbothsides.}}\hfill \\ P-b& =& 2a& \\ \frac{P-b}{{2}}& =& \frac{2a}{{2}}& \mathit{\text{Dividebothsidesby2.}}\hfill \\ \frac{P-b}{2}& =& a& \\ & & & \end{array}$$ Solution: $a=\frac{P-b}{2}$
Solve for x: $z=\frac{x+y}{2}$ $$\begin{array}{cccc}z& =& \frac{x+y}{2}& \\ {2}{\cdot}z& =& {2}{\cdot}\frac{x+y}{2}& \mathit{\text{Multiplybothsidesby2}}\text{.}\hfill \\ 2z& =& x+y& \\ 2z{-}{y}& =& x+y{-}{y}& \mathit{\text{Subtractyonbothsides}}\text{.}\hfill \\ 2z-y& =& x& \end{array}$$ Solution $x=2z-y$
$\text{Is}x=7\text{a solution to}-3x+5=-16\text{?}$
Yes
$\text{Is}x=2\text{asolutionto}-2x-7=28\text{?}$
No
$\text{Is}x=-3\text{asolutionto}\frac{1}{3}x-4=-5\text{?}$
Yes
$\text{Is}x=-2\text{asolutionto}3x-5=-2x-15\text{?}$
Yes
$\text{Is}x=-\frac{1}{2}\text{asolutionto}3(2x+1)=-4x-3\text{?}$
No
$\text{Solveforx:}x-5=-8$
$x=-3$
$\text{Solvefory:}-4+y=-9$
$y=-5$
$\text{Solveforx:}x-\frac{1}{2}=\frac{1}{3}$
$x=\frac{5}{6}$
$\text{Solveforx:}x+2{\scriptscriptstyle \frac{1}{2}}=3{\scriptscriptstyle \frac{1}{3}}$
$x=\frac{5}{6}$
$\text{Solveforx:}4x=-44$
$x=-11$
$\text{Solvefora:}-3a=-30$
$a=10$
$\text{Solvefory:}27=9y$
$y=3$
$\text{Solveforx:}\frac{x}{3}=-\frac{1}{2}$
$x=-\frac{3}{2}$
$\text{Solvefort:}-\frac{t}{12}=\frac{1}{4}$
$t=-3$
$\text{Solveforx:}\frac{7}{3}x=-\frac{1}{2}$
$x=-\frac{3}{14}$
$\text{Solvefora:}3a-7=23$
$a=10$
$\text{Solvefory:}-3y+2=-13$
$y=5$
$\text{Solveforx:}-5x+8=8$
$x=0$
$\text{Solveforx:}\frac{1}{2}x+\frac{1}{3}=\frac{2}{5}$
$x=\frac{2}{15}$
$\text{Solvefory:}3-2y=-11$
$y=7$
$\text{Solveforx:}-10=2x-5$
$x=-\frac{5}{2}$
$\text{Solvefora:}4a-\frac{2}{3}=-\frac{1}{6}$
$a=\frac{1}{8}$
$\text{Solveforx:}\frac{\text{3}}{\text{5}}x-\frac{1}{2}=\frac{1}{10}$
$x=1$
$\text{Solvefory:}-\frac{4}{5}y+\frac{1}{3}=\frac{1}{15}$
$y=\frac{1}{3}$
$\text{Solveforx:}-x-5=-2$
$x=-3$
$\text{Solveforx:}3x-5=2x-17$
$x=-12$
$\text{Solvefory:}-2y-7=3y+13$
$y=-4$
$\text{Solvefora:}\frac{1}{2}a-\frac{2}{3}=a+\frac{1}{5}$
$a=-\frac{26}{15}$
$\text{Solveforx:}-2+4x+9=7x+8-2x$
$x=-1$
$\text{Solvefora:}3a+5-x=2a+7$
No Solution, $\varnothing $
$\text{Solveforb:}-7b+3=2-5b+1-2b$
All Reals, $\Re $
$\text{Solvefory:}-5(2y-3)+2=12$
$y=\frac{1}{2}$
$\text{Solveforx:}3-2(x+4)=-3(4x-5)$
$x=2$
$\text{Solvefora:}-3(2a-3)+2=3(a+7)$
$a=-\frac{10}{9}$
$\text{Solveforx:}10(3x+5)-5(4x+2)=2(5x+20)$
All Reals, $\Re $
$\text{Solveforw:}P=2l+2w$
$w=\frac{P-2l}{2}$
$\text{Solveforb:}P=a+b+c$
$b=P-a-c$
$\text{SolveforC:}F=\frac{9}{5}C+32$
$C=\frac{5F-160}{9}$
$\text{Solveforr:}C=2\pi r$
$r=\frac{C}{2\pi}$
$\text{Solvefory:}z=\frac{x-y}{5}$
$y=-5z+x$
Notification Switch
Would you like to follow the 'Contemporary math applications' conversation and receive update notifications?