# 5.2 Solve systems of equations by substitution  (Page 4/5)

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The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

The length is 12 and the width is 8.

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

The length is 23 and the width is 6.

For [link] we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

## Solution

We will draw and label a figure.

 Step 1. Read the problem. Step 2. Identify what you are looking for. We are looking for the measures of the angles. Step 3. Name what we are looking for. Let $a=$ the measure of the 1 st angle $\phantom{\rule{1.5em}{0ex}}b=$ the measure of the 2 nd angle Step 4. Translate into a system of equations. The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. The sum of the measures of the angles of a triangle is 180. The system is: Step 5. Solve the system of equations. We will use substitution since the first equation is solved for a . Substitute 3 b + 10 for a in the second equation. Solve for b . Substitute b = 20 into the first equation and then solve for a . Step 6. Check the answer in the problem. We will leave this to you! Step 7. Answer the question. The measures of the small angles are 20 and 70.

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

The measure of the angles are 22 degrees and 68 degrees.

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

The measure of the angles are 36 degrees and 54 degrees.

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus$15 for each training session. Option B would pay her $10,000 +$40 for each training session. How many training sessions would make the salary options equal?

## Solution

 Step 1. Read the problem. Step 2. Identify what you are looking for. We are looking for the number of training sessions that would make the pay equal. Step 3. Name what we are looking for. Let $s=$ Heather’s salary. $\phantom{\rule{1.5em}{0ex}}n=$ the number of training sessions Step 4. Translate into a system of equations. Option A would pay her $25,000 plus$15 for each training session. Option B would pay her $10,000 +$40 for each training session The system is: Step 5. Solve the system of equations. We will use substitution. Substitute 25,000 + 15 n for s in the second equation. Solve for n . Step 6. Check the answer. Are 600 training sessions a year reasonable? Are the two options equal when n = 600? Step 7. Answer the question. The salary options would be equal for 600 training sessions.

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of$100 for each policy sold. The second pays a salary of $20,000 plus a commission of$50 for each policy sold. How many policies would need to be sold to make the total pay the same?

There would need to be 160 policies sold to make the total pay the same.

#### Questions & Answers

A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Mikaela Reply
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?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
Nakiya Reply
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than \$110,000
Michael

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