# 5.3 Solve systems of equations by elimination  (Page 4/6)

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The sum of two numbers is 39. Their difference is 9. Find the numbers.

## Solution

$\begin{array}{ccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem}\hfill & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & \hfill \text{We are looking for two numbers.}\hfill \\ \mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & \begin{array}{c}\hfill \text{Let}\phantom{\rule{0.2em}{0ex}}n=\phantom{\rule{0.2em}{0ex}}\text{the first number.}\hfill \\ \hfill \phantom{\rule{2.3em}{0ex}}m=\text{the second number}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{into a system of equations.}\hfill \\ \\ \\ \\ \\ \\ \\ \end{array}\hfill & & \hfill \begin{array}{}\\ \text{The sum of two numbers is 39.}\hfill \\ \hfill \phantom{\rule{0.3em}{0ex}}n+m=39\hfill \\ \hfill \text{Their difference is 9.}\hfill \\ \hfill n-m=9\hfill \end{array}\hfill \\ \text{The system is:}\hfill & & \hfill \left\{\begin{array}{c}n+m=39\hfill \\ n-m=9\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the system of equations.}\hfill \\ \text{To solve the system of equations, use}\hfill \\ \text{elimination. The equations are in standard}\hfill \\ \text{form and the coefficients of}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{are}\hfill \\ \text{opposites. Add.}\hfill \\ \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}n.\hfill \\ \\ \\ \\ \\ \end{array}\hfill & & \hfill \begin{array}{c}\hfill \underset{\text{____________}}{\left\{\begin{array}{c}n+m=39\hfill \\ n-m=9\hfill \end{array}}\hfill \\ \hfill 2n\phantom{\rule{1.8em}{0ex}}=48\hfill \\ \\ \hfill \phantom{\rule{2.21em}{0ex}}n=24\hfill \end{array}\hfill \\ \begin{array}{c}\text{Substitute}\phantom{\rule{0.2em}{0ex}}n=24\phantom{\rule{0.2em}{0ex}}\text{into one of the original}\hfill \\ \text{equations and solve for}\phantom{\rule{0.2em}{0ex}}m.\hfill \end{array}\hfill & & \hfill \begin{array}{c}\hfill \phantom{\rule{0.2em}{0ex}}\begin{array}{c}\hfill n+m=39\\ \hfill 24+m=39\\ \hfill m=15\end{array}\hfill \end{array}\hfill \\ \mathbf{\text{Step 6. Check}}\phantom{\rule{0.2em}{0ex}}\text{the answer.}\hfill & & \phantom{\rule{1.55em}{0ex}}\text{Since}\phantom{\rule{0.2em}{0ex}}24+15=39\phantom{\rule{0.2em}{0ex}}\text{and}\hfill \\ & & \hfill 24-15=9,\phantom{\rule{0.2em}{0ex}}\text{the answers check.}\hfill \\ \mathbf{\text{Step 7. Answer}}\phantom{\rule{0.2em}{0ex}}\text{the question.}\hfill & & \hfill \text{The numbers are 24 and 15.}\hfill \end{array}$

The sum of two numbers is 42. Their difference is 8. Find the numbers.

The numbers are 25 and 17.

The sum of two numbers is −15. Their difference is −35. Find the numbers.

The numbers are −25 and 10.

Joe stops at a burger restaurant every day on his way to work. Monday he had one order of medium fries and two small sodas, which had a total of 620 calories. Tuesday he had two orders of medium fries and one small soda, for a total of 820 calories. How many calories are there in one order of medium fries? How many calories in one small soda?

## Solution

 Step 1. Read the problem. Step 2. Identify what we are looking for. We are looking for the number of calories in one order of medium fries and in one small soda. Step 3. Name what we are looking for. Let f = the number of calories in 1 order of medium fries.     s = the number of calories in 1 small soda. Step 4. Translate into a system of equations: one medium fries and two small sodas had a total of 620 calories two medium fries and one small soda had a total of 820 calories. Our system is: Step 5. Solve the system of equations. To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of f , multiply the top equation by −2. Simplify and add. Solve for s . Substitute s = 140 into one of the original equations and then solve for f . Step 6. Check the answer. Verify that these numbers make sense in the problem and that they are solutions to both equations. We leave this to you! Step 7. Answer the question. The small soda has 140 calories and the fries have 340 calories.

Malik stops at the grocery store to buy a bag of diapers and 2 cans of formula. He spends a total of $37. The next week he stops and buys 2 bags of diapers and 5 cans of formula for a total of$87. How much does a bag of diapers cost? How much is one can of formula?

The bag of diapers costs $11 and the can of formula costs$13.

To get her daily intake of fruit for the day, Sasha eats a banana and 8 strawberries on Wednesday for a calorie count of 145. On the following Wednesday, she eats two bananas and 5 strawberries for a total of 235 calories for the fruit. How many calories are there in a banana? How many calories are in a strawberry?

There are 105 calories in a banana and 5 calories in a strawberry.

A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs$9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot? Parker Reply what you wanna get Cesar 800 sq. ft @$1.50 & 200 sq. ft @ $9.00 Marco Geneva treated her parents to dinner at their favorite restaurant. The bill was$74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 =$11.88 total bill: $74.25 +$11.88 = $86.13 ericka yes and tip 16% will be$11.88
David
what is the shorter way to do it