Math 0530/1530

4.4 Solving Equations

Rachel is planting a new dogwood tree in her yard. The tree is 12 feet tall now and expected to grow another foot every year.
1. How tall will the tree be one year from now? 13 feet
2. How tall will the tree be five years from now? 17 feet
3. How tall will the tree be 15 years from now? 27 feet
4. Use a variable to represent the number of years from now, and write an expression to represent the height of the tree. $12+y$
5. How many years after planting the tree will Rachel have to wait before the tree is 19.75 feet tall?
  The tree will be 19.75 feet tall in 7.75 years.
A certain credit card company is offering its customers 5% cash back on all purchases made using their card during a calendar year.
$0.05x$
1. If the amount of purchases you made using the card for the year is $24,000, how much will the company give back to you?
2. If you purchase fifteen thousand dollars of items using the card one year, what will the company give back to you at the end of the year?
3. If you have received $60.00 back from the credit card company, what was the total amount of purchases you made using the card for the year?
4. If the company sent you a check for $1826.20, what is the total amount of purchases you made for the year on their card?
Julia feels like she needs to spring clean her house, but she also works full time so the time to complete her cleaning is limited. She has 25 items on her very detailed list of tasks and plans to accomplish one item on the list each day for the next few weeks.
1. After 3 days, how many tasks does Julia have remaining on her list?
  22 tasks
2. Julia has been working on her list for 16 days. How many tasks does she still have on her list?
  9 tasks
3. After one week, how many tasks are still on Julia’s list?
  18 tasks
4. Choose a variable to represent the amount of time that Julia has been working on completing the items on her list and use this variable to write a rule for the number of tasks remaining on her list.
  $25-d$
5. If Julia has only 6 items left on her list, how many days has she been working on her spring cleaning?
  $25-d =6$
  
  $25-d-25=6-25$
  
  $-d=-19$
  
  $\frac{-d}{-1}=\frac{-19}{-1}$
  
  19 days

Solve for the variable. Show all your steps.

$x+2=-9$

$x+2-2=-9-2$

$x=-11$

$\frac{w}{2}=-4.4$

$\frac w{2}\left(2\right)=-4.4\left(2\right)$

$a=-8.8$

$15 x=-225$

$\frac{15x}{15}=\frac{-225}{15}$

$w=-15$

$a-8=-7$

$a-8+8=-7+8$

$a=1$

$-4x=-18$

$\frac{-4x}{-4}=\frac{-18}{-4}$

$x=4.5$

$x-20.74=83.9$

$x-20.74+20.74=83.9+20.74$

$x=104.64$

$-x=-16$

$\frac{-x}{-1}=\frac{-16}{-1}$

$x=16$

$-62.72=-3.92x$

$\frac{-62.72}{-3.92}=\frac{-3.92x}{-3.92}$

$16=x$

$-x+16=-10$

$-x+16-16=-10-16$

$-x=-26$

$\frac{-x}{-1}=\frac{-26}{-1}$

$x=26$

$15-x=1$

$15-x-15=1-15$

$-x=-14$

$\frac{-x}{-1}=\frac{-14}{-1}$

$x=14$

Daniel and his wife need to purchase a new water heater for their home. They have chosen to purchase a 65-gallon heater because of the size of their family. The water heater will cost $700. The cost for them to have the old water heater removed, the area prepared for the new one and the new water heater installed will cost them $90 per hour.
If Daniel is invoiced for $1285.00, how long did the plumber work?

$90x+700=1285$

$90x+700-700=1285-700$

$90x=585$

$\frac{90x}{90}=\frac{\displaystyle585}{90}$

$x=6.5$

The plumber worked 6.5 hours.
Phyllis has been saving money for a long time to take a vacation with her daughter. After many months, she has saved $1875. She wants to spend her money wisely to make it stretch throughout the entire vacation, so she plans to spend $150 per day.
1. If they spend $1500, how long is their vacation?
  $1875-150d=375$
  
  $1875-150d-1875=375-1875$
  
  $-150d=-1500$
  
  $\frac{-150D}{-150}=\frac{\displaystyle-1500}{-150}$
  
  $d=10$
  
  Their vacation was 10 days long.
2. If they have no money left when they return and they stick to their $150 per day budget, how many days is their vacation?
  $1875-150d=0$
  
  $1875-150d-1875=0-1875$
  
  $-150d=-1875$
  
  $\frac{-150D}{-150}=\frac{\displaystyle-1875}{-150}$
  
  $d=12.5$
  
  Their vacation was 12.5 days long.
Jim is concerned that the grass he cuts is growing too slowly and he will not be mowing enough yards to earn school money. He finds that the grass measures 2 inches when he cuts it and grows at a rate of 0.125 inches per day.
1. What will be the height of the grass 2 weeks after it is cut?
  $0.125(14)+2$
  
  $1.75+2$
  
  $3.75$
  
  The grass will be 3.75 inches tall.
2. In how many days after it is cut will the grass be 3 inches tall?
  $0.125d+2=3$
  
  $0.125d+2-2=3-2$
  
  $0.125d=1$
  
  $\frac{0.125d}{0.125}=\frac1{0.125}$
  
  $d=8$
  
  The grass will be 3 inches tall 8 days after it is cut.
3. In how many weeks after it is cut will the grass be 1 foot?
  $0.125d+2=12$
  
  $0.125d+2-2=12-2$
  
  $0.125d=10$
  
  $\frac{0.125d}{0.125}=\frac{10}{0.125}$
  
  $d=80$
  
  The grass will be 1 foot tall 80 days after it is cut.

Solve for the variable. Show all your steps.

$-9 x+1=-80$

$-9x+1-1=-80-1$

$-9x=-81$

$\frac{-9x}{-9}=\frac{-81}{-9}$

$m=9$

$8 n-7=25$

$8n-7+7=25+7$

$8n=32$

$\frac{8n}{8}=\frac{32}{8}$

$n=4$

$10-6 v=-104$

$10-6 v-10=-104-10$

$-6 v=-114$

$\frac{-6v}{-6}=\frac{-114}{-6}$

$v=19$

$-10=-10+7 m$

$-10+10=-10+7 m+10$

$0=7 m$

$\frac{0}{7}=\frac{7m}{7}$

$m=0$

$-103=-13-9x$

$-103+13=-13+13-9x$

$-90=-9x$

$\frac{-90}{-9}=\frac{-9x}{-9}$

$x=10$

$498.52=14.3+6.2x$

$498.52-14.3=14.3-14.3+6.2x$

$484.22=6.2x$

$\frac{484.22}{6.2}=\frac{6.2x}{6.2}$

$x=78.1$

Solve for x:

$ax+b=c$

$ax+b-b=c-b$

$ax=c-b$

$\frac{ax}a=\frac{c-b}a$

$x=\frac{c-b}a$