# 2.3 Solve equations with variables and constants on both sides  (Page 3/3)

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Solve: $2a-2=6a+18.$

$a=-5$

Solve: $4k-1=7k+17.$

$k=-6$

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

Solve: $\frac{5}{4}x+6=\frac{1}{4}x-2.$

## Solution

Since $\frac{5}{4}>\frac{1}{4}$ , make the left side the “variable” side and the right side the “constant” side.

 Subtract $\frac{1}{4}x$ from both sides. Combine like terms. Subtract $6$ from both sides. Simplify. $\begin{array}{cccccc}\text{Check:}\hfill & & & \hfill \frac{5}{4}x+6& =\hfill & \frac{1}{4}x-2\hfill \\ \text{Let}\phantom{\rule{0.2em}{0ex}}x=-8.\hfill & & & \hfill \frac{5}{4}\left(-8\right)+6& \stackrel{?}{=}\hfill & \frac{1}{4}\left(-8\right)-2\hfill \\ & & & \hfill -10+6& \stackrel{?}{=}\hfill & -2-2\hfill \\ & & & \hfill -4& =\hfill & -4✓\hfill \end{array}$

Solve: $\frac{7}{8}x-12=-\frac{1}{8}x-2.$

$x=10$

Solve: $\frac{7}{6}y+11=\frac{1}{6}y+8.$

$y=-3$

We will use the same strategy to find the solution for an equation with decimals.

Solve: $7.8x+4=5.4x-8.$

## Solution

Since $7.8>5.4$ , make the left side the “variable” side and the right side the “constant” side.

 Subtract $5.4x$ from both sides. Combine like terms. Subtract $4$ from both sides. Simplify. Use the Division Propery of Equality. Simplify. Check: Let $x=-5$ .

Solve: $2.8x+12=-1.4x-9.$

$x=-5$

Solve: $3.6y+8=1.2y-4.$

$y=-5$

## Key concepts

• Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation
1. Choose which side will be the “variable” side—the other side will be the “constant” side.
2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
5. Check the solution by substituting it into the original equation.

## Practice makes perfect

Solve Equations with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

$9x-3=60$

$12x-8=64$

$x=6$

$14w+5=117$

$15y+7=97$

$y=6$

$2a+8=-28$

$3m+9=-15$

$m=-8$

$-62=8n-6$

$-77=9b-5$

$b=-8$

$35=-13y+9$

$60=-21x-24$

$x=-4$

$-12p-9=9$

$-14q-2=16$

$q=-\frac{9}{7}$

Solve Equations with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

$19z=18z-7$

$21k=20k-11$

$k=-11$

$9x+36=15x$

$8x+27=11x$

$x=9$

$c=-3c-20$

$b=-4b-15$

$b=-3$

$9q=44-2q$

$5z=39-8z$

$z=3$

$6y+\frac{1}{2}=5y$

$4x+\frac{3}{4}=3x$

$x=-\frac{3}{4}$

$-18a-8=-22a$

$-11r-8=-7r$

$r=-2$

Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

$8x-15=7x+3$

$6x-17=5x+2$

$x=19$

$26+13d=14d+11$

$21+18f=19f+14$

$f=7$

$2p-1=4p-33$

$12q-5=9q-20$

$q=-5$

$4a+5=\text{−}a-40$

$8c+7=-3c-37$

$c=-4$

$5y-30=-5y+30$

$7x-17=-8x+13$

$x=2$

$7s+12=5+4s$

$9p+14=6+4p$

$p=-\frac{8}{5}$

$2z-6=23-z$

$3y-4=12-y$

$y=4$

$\frac{5}{3}c-3=\frac{2}{3}c-16$

$\frac{7}{4}m-7=\frac{3}{4}m-13$

$m=-6$

$8-\frac{2}{5}q=\frac{3}{5}q+6$

$11-\frac{1}{5}a=\frac{4}{5}a+4$

$a=7$

$\frac{4}{3}n+9=\frac{1}{3}n-9$

$\frac{5}{4}a+15=\frac{3}{4}a-5$

$a=-40$

$\frac{1}{4}y+7=\frac{3}{4}y-3$

$\frac{3}{5}p+2=\frac{4}{5}p-1$

$p=15$

$14n+8.25=9n+19.60$

$13z+6.45=8z+23.75$

$z=3.46$

$2.4w-100=0.8w+28$

$2.7w-80=1.2w+10$

$w=60$

$5.6r+13.1=3.5r+57.2$

$6.6x-18.9=3.4x+54.7$

$x=23$

## Everyday math

Concert tickets At a school concert the total value of tickets sold was $1506. Student tickets sold for$6 and adult tickets sold for $9. The number of adult tickets sold was 5 less than 3 times the number of student tickets. Find the number of student tickets sold, s , by solving the equation $6s+27s-45=1506$ . Making a fence Jovani has 150 feet of fencing to make a rectangular garden in his backyard. He wants the length to be 15 feet more than the width. Find the width, w , by solving the equation $150=2w+30+2w$ . 30 feet ## Writing exercises Solve the equation $\frac{6}{5}y-8=\frac{1}{5}y+7$ explaining all the steps of your solution as in the examples in this section. Solve the equation $10x+14=-2x+38$ explaining all the steps of your solution as in the examples in this section. $x=2$ Justifications will vary. When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient of $x$ to be the “variable” side? Is $x=-2$ a solution to the equation $5-2x=-4x+1$ ? How do you know? Yes. Justifications will vary. ## Self check After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. What does this checklist tell you about your mastery of this section? What steps will you take to improve? #### Questions & Answers integer greater than 2 and less than 12 Emily Reply 2 < x < 12 Felix I'm guessing you are doing inequalities... Felix Actually, translating words into algebraic expressions / equations... Felix hi Darianna hello Mister He charges$125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least$2,400?
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
they are 92 candies in the bag
POP
rectangular field solutions
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif how muqtaar Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif -5p+30? Corey p=-5+30 Jacob How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando No; 65m/hr albert hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce