Equations with parentheses - Questions

Answers

1.

We need to manipulate the equation to get \(c\) by itself.

\(4(3+c)+c=c+4\)
\( 12+4 c+c =c+4 \) Distribute.
\( 5 c+12 =c+4 \) Combine like terms.
\( 5 c+12-c =c+4-c \) Subtract \(c\) from each side.
\( 4 c+12 =4 \) Combine like terms.
\( 4 c+12-12 =4-12 \) Subtract \(12\) from each side.
\( 4 c =-8 \) Combine like terms.
\( \frac{4 c}{4} =\frac{-8}{4} \) Divide each side by \(4\).
\( c =-2 \) Simplify.


The answer: \(c = -2\)


Let's check our work!

\(\begin{gathered}

4(3+c)+c=c+4 \\

4(3+(-2))+(-2) \stackrel{?}{=}-2+4 \\

4(1)-2 \stackrel{?}{=} 2 \\

4-2 \stackrel{?}{=} 2 \\

2=2 \quad \text { Yes! }

\end{gathered}\)


2. \(p = 10\)

We need to manipulate the equation to get \(p\) by itself.

\(10p−3=2(12+4p)−7\)
\( 10 p-3 =24+8 p-7 \) Distribute.
\( 10 p-3 =17+8 p \) Combine like terms.
\( 10 p-3-8 p =17+8 p-8 p\) Subtract \(8p\) from each side.
\( 2 p-3 =17 \) Combine like terms.
\( 2 p-3+3 =17+3 \) Add \(3\) to each side.
\( 2 p =20 \) Combine like terms.
\( \frac{2 p}{2} =\frac{20}{2} \) Divide each side by \(2\).
\( p =10 \) Simplify.


The answer: \(p = 10\)


Let's check our work!

\(\begin{aligned}

10 p-3 &=2(12+4 p)-7 \\

10(10)-3 & \stackrel{?}{=} 2(12+4(10))-7 \\

100-3 & \stackrel{?}{=} 2(12+40)-7 \\

97 & \stackrel{?}{=} 2(52)-7 \\

97 & \stackrel{?}{=} 104-7 \\

97 &=97 \quad \text { Yes! }

\end{aligned}\)


3. \(t = 9\)

We need to manipulate the equation to get \(t\) by itself.

\(−t=9(t−10)\)
\( -t =9 t-90 \) Distribute.
\( -t-9 t =9 t-90-9t \) Subtract \(9t\) from each side.
\( -10 t =-90 \) Combine like terms.
\( \frac{-10 t}{-10} =\frac{-90}{-10} \) Divide each side by \(-10\).
\( t =9 \) Simplify.


The answer is: \(t = 9\)


Let's check our work!

\(\begin{aligned}

&-t=9(t-10) \\

&-9 \stackrel{?}{=} 9(9-10) \\

&-9 \stackrel{?}{=} 9(-1) \\

&-9=-9 \quad \text { Yes! }

\end{aligned}\)


4. \(g = 4\)

We need to manipulate the equation to get \(g\) by itself.

\(6(−2g−1)=−(13g+2)\)
\( -12 g-6 =-13 g-2 \) Distribute.
\( -12 g-6+12 g =-13 g+12 g-2 \) Add \(12g\) to each side.
\( -6+2 =-g-2+2 \) Add \(2\) to each side.
\( -4 =-g \) Divide by \(-1\).
\( 4 =g \) Combine like terms.


The answer: \(g = 4\)


Let's check our work!

\(\begin{aligned}

6(-2 g-1) &=-(13 g+2) \\

6(-2(4)-1) & \stackrel{?}{=}-(13(4)+2) \\

6(-8-1) & \stackrel{?}{=}-(52+2) \\

6(-9) & \stackrel{?}{=}-(54) \\

-54 &=-54 \quad \text { Yes! }

\end{aligned}\)