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Equations with variables on both sides

Site: Saylor Academy
Course: GKT101: General Knowledge for Teachers – Math
Book: Equations with variables on both sides
Printed by: Guest user
Date: Sunday, May 19, 2024, 10:21 PM

Description

This lecture series shows how you can apply the principle of doing the same thing to both sides of the equation to equations with variables on both sides. Watch the videos and complete the interactive exercise sets.

Table of contents

Why we do the same thing to both sides: Variable on both sides

Source: Khan Academy, https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq#alg-variables-on-both-sides
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Intro to equations with variables on both sides

Equations with variables on both sides: 20-7x=6x-6

Equation with variables on both sides: fractions

Equation with the variable in the denominator

Figuring out missing algebraic step

Equations with variables on both sides - Questions

1. Solve for e.

9e−7=7e−11


2. Solve for p.

17−2p=2p+5+2p


3. Solve for a.

5+14a=9a−5


4. Solve for m.

−7+4m+10=15−2m

Answers

1. e=−2

We need to manipulate the equation to get e by itself.

9e−7=7e−11
9 e-7-7 e=7 e-11-7 e Subtract 7e from each side.
2 e-7=-11 Combine like terms.
2 e-7+7=-11+7 Add 7 to each side.
2 e=-4 Combine like terms.
\frac{2 e}{2}=\frac{-4}{2} Divide each side by 2.
e=-2 Simplify,


The answer: e=−2


Let's check our work!

\begin{gathered} 9 e-7=7 e-11 \\ 9(-2)-7 \stackrel{?}{=} 7(-2)-11 \\ -18-7 \stackrel{?}{=}-14-11 \\ -25=-25 \quad \text { Yes! } \end{gathered}


2. p = 2

We need to manipulate the equation to get p by itself.

17−2p=2p+5+2p
17-2 p =4 p+5 Combine like terms.
17-2 p-4 p =4 p+5-4 p Subtract 4p from each side.
-6 p+17 =5 Combine like terms.
-6 p+17-17 =5-17 Subtract 17 from each side.
\frac{-6 p}{-6 p} =-12 Combine like terms.
\frac{-6}{-6} =\frac{-12}{-6} Divide each side by -6
p = 2 Simplify.


The answer: p = 2


Let's check our work!

\begin{aligned} 17-2 p &=2 p+5+2 p \\ 17-2 p &=4 p+5 \\ 17-2(2) & \stackrel{?}{=} 4(2)+5 \\ 17-4 & \stackrel{?}{=} 8+5 \\ 13 &=13 \quad \text { Yes! } \end{aligned}


3. a =-2

We need to manipulate the equation to get a by itself.

5+14a=9a−5
5+14 a-9 a =9 a-5-9 a Subtract 9a from each side.
5+5 a =-5 Combine like terms.
5+5 a-5 =-5-5 Subtract 5 from each side.
5 a =-10 Combine like terms.
\frac{5 a}{5} =\frac{-10}{5} Divide each side by 5.
a =-2 Simplify.


The answer: a =-2


Let's check our work!

\begin{aligned} 5+14 a &=9 a-5 \\ 5+14(-2) & \stackrel{?}{=} 9(-2)-5 \\ 5+(-28) & \stackrel{?}{=}-18-5 \\ -23 &=-23 \quad \text { Yes! } \end{aligned}


4. m = 2

We need to manipulate the equation to get m by itself.

−7+4m+10=15−2m
4 m+3 =15-2 m Combine like terms.
3+4 m+2 m =15-2 m+2 m Add 2m to each side.
6 m+3 =15 Combine like terms.
6 m+3-3 =15-3 Subtract 3 from each side.
6 m =12 Combine like terms.
\frac{6 m}{6} =\frac{12}{6} Divide each side by 6.
m = 2 Simplify.


The answer: m = 2


Let's check our work!

\begin{aligned} -7+4 m+10 &=15-2 m \\ 3+4 m &=15-2 m \\ 3+4(2) & \stackrel{?}{=} 15-2(2) \\ 3+8 & \stackrel{?}{=} 15-4 \\ 11 &=11 \quad \text { Yes! } \end{aligned}

Equations with variables on both sides: decimals & fractions - Questions

1. Solve for k.

4.5+1.5k=18−3k


2. Solve for s.

2-2 s=\frac{3}{4} s+13


3. Solve for g.

9+3.5g=11−0.5g


4. Solve for p.

16-3 p=\frac{2}{3} p+5

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}