Equations with variables on both sides
Equations with variables on both sides: decimals & fractions - Questions
Answers
1. \(k = 3\)
We need to manipulate the equation to get \(k\) by itself.
| \(4.5+1.5k=18−3k\) | |
| \( 4.5+1.5 k+3 k =18-3 k+3 k \) | Add \(3k\) to each side. |
| \( 4.5 k+4.5 =18 \) | Combine like terms. |
| \( 4.5 k+4.5-4.5 =18-4.5 \) | Subtract \(4.5\) from each side. |
| \( 4.5 k =13.5 \) | Combine like terms. |
| \( \frac{4.5 k}{4.5} =\frac{13.5}{4.5} \) | Divide each side by \(4.5\). |
| \( k =3 \) | Simplify. |
The answer: \(k = 3\)
Let's check our work!
\(\begin{aligned}
4.5+1.5 k &=18-3 k \\
4.5+1.5(3) & \stackrel{?}{=} 18-3(3) \\
4.5+4.5 & \stackrel{?}{=} 18-9 \\
9 &=9 \quad \text { Yes! }
\end{aligned}\)
2.
We need to manipulate the equation to get \(s\) by itself.
| \(2-2 s=\frac{3}{4} s+13\) | |
| \( 2-2 s-\frac{3}{4} s =\frac{3}{4} s+13-\frac{3}{4} s\) | Subtract \(\frac{3}{4}\) from each side. |
| \( -\frac{11}{4} s+2 =13 \) | Combine like terms. |
| \( -\frac{11}{4} s+2-2 =13-2 \) | Subtract \(2\) from each side. |
| \( -\frac{11}{4} s =11 \) | Combine like terms. |
| \( s \cdot\left(-\frac{4}{11}\right) =11 \cdot\left(-\frac{4}{11}\right)\) | Multiply each side by \(-\frac{4}{11}\). |
| \( s =-\frac{44}{11} \) | |
| \( s =-4 \) | Simplify. |
The answer: \(s = -4\)
Let's check our work!
\(\begin{aligned}
2-2 s &=\frac{3}{4} s+13 \\
2-2(-4) & \stackrel{?}{=} \frac{3}{4}(-4)+13 \\
2+8 & \stackrel{?}{=}-\frac{12}{4}+13 \\
10 & \stackrel{?}{=}-3+13 \\
10 &=10 \quad \text { Yes! }
\end{aligned}\)
3. \(g = 0.5\)
We need to manipulate the equation to get \(g\) by itself.
| \(9+3.5g=11−0.5g\) | |
| \( 9+3.5 g+0.5 g =11-0.5 g+0.5 g \) | Add \(0.5g\) to each side. |
| \( 9+4 g =11 \) | Combine like terms. |
| \( 4 g+9-9 =11-9 \) | Subtract \(9\) from each side. |
| \( 4 g =2 \) | Combine like terms. |
| \( \frac{4 g}{4} =\frac{2}{4} \) | Divide each side by \(4\). |
| \( g =0.5 \) | Simplify. |
The answer: \(g = 0.5\)
Let's check our work!
\(\begin{aligned}
9+3.5 g &=11-0.5 g \\
9+3.5(0.5) & \stackrel{?}{=} 11-0.5(0.5) \\
9+1.75 & \stackrel{?}{=} 11-0.25 \\
10.75 &=10.75 \quad \text { Yes! }
\end{aligned}\)
4. \(p = 3\)
We need to manipulate the equation to get \(p\) by itself.
| \(16-3 p=\frac{2}{3} p+5\) | |
| \( 16-3 p-\frac{2}{3} p =\frac{2}{3} p+5-\frac{2}{3} p \) | Subtract \(\frac{2}{3}p\) from each side. |
| \( -\frac{11}{3} p+16 =5 \) | Combine like terms. |
| \( -\frac{11}{3} p+16-16 =5-16 \) | Subtract \(16\) from each side. |
| \( -\frac{11}{3} p =-11 \) | Combine like terms. |
| \( -\frac{11}{3} p \cdot\left(-\frac{3}{11}\right) =-11 \cdot\left(-\frac{3}{11}\right) \) | Multiply each side by \(-\frac{3}{11}\) |
| \( p =\frac{33}{11}\) | |
| \( p =3 \) | Simplify. |
The answer: \(p = 3\)
Let's check our work!
\(\begin{aligned}
16-3 p &=\frac{2}{3} p+5 \\
16-3(3) & \stackrel{?}{=} \frac{2}{3}(3)+5 \\
16-9 & \stackrel{?}{=} 2+5 \\
7 &=7 \quad \text { Yes! }
\end{aligned}\)