So far in this unit, you have seen several examples of problems that can be solved by using systems. We are going to review how we set them up here.
Example 1: Phone Costs from Section 1, Part 3
Set up the following problem using two equations and the variables, x and y, in each.
Sara wants to sign up for a telephone service. The cost of a local phone line is $36 per month, plus $0.10 per long distance call. The cost of a cell phone is $50 per month with unlimited long distance, as long as the minutes do not go over 1,000. How many long distance calls would Sara need to make on the local phone line to spend as much as she would if she used a cell phone?
Find the linear equations.
Phone
Monthly
per minute charge
Equation y = mx + b
Telephone
$36
$0.10
y = 36 + 0.10x
Cell Phone
$50
no charge
y = 50
You may go back to the section and part listed to see how we graphed this to find the common solution.
We found the ordered pair (140, 50).
The solution is that if Sara talked 140 long distance minutes per month, the cost of both phones would be the same, $50.
Example 2: Dryer Costs from Section 1, Part 4
Set up the following problem using two equations and the variables x and y in each.
The electric clothes dryer you are interested in buying costs $300 with tax. It uses $0.50 of electricity for one hour of drying. It costs $1.50 per hour to dry clothes at a laundromat. When would the cost of buying and using the dryer at home be the same as going to the laundromat?
Write the two equations in y = mx + b form:
Dryer costs at home: y = 300 + 0.50x
Dryer costs at the laundromat: y = 0 + 1.50x
Where x is the number of loads taking an hour to dry and y is the total cost.
You can review how they were graphed in the section listed above to find the common solution.
The solution was (300, 450).
If you do 300 loads of laundry, the cost of using the dryer at home and the dryer in the laundromat is the same, $450.
Example 3: Appliance Sales from Section 2, Part 1
Set up the following problem using two equations and the variables x and y in each.
George is considering two different job offers. Both jobs involve selling appliances. In one store, he would make $183 per week plus a 5% commission on the appliances he sold. In the other store, he would get $240 per week with a 2.5% commission on appliances sold. How much would George have to sell before he would make the same at either job?
Use y = mx + b.
Set up the two equations by letting
x = total price of appliances sold
and y = his weekly pay.
For the first job: y = .05x + 183
For the second: y = .025x + 240
This was solved using the substitution method.
The solution was (2280, 297).
George would have to sell $2,280 worth of appliances to make the same on both jobs, $297.
Example 4: Puzzle Problem from Section 2, Part 2
Set up the following problem using two equations and the variables, x and y, in each.
Here is a number puzzle. Two numbers have a sum of 40. Their difference is 16. What are the two numbers?
Substitution
This time, just directly interpret the equation.
Let the two numbers be m and n.
m + n = 40
m – n = 16
This was solved using elimination.
The solution was (28, 12).
The two numbers are 28 and 12.
Example 5: Cost of Eating Out from Section 2, Part 3
Set up the following problem using two equations and the variables, x and y, in each.
To celebrate a win for the academic team, the coach treated them to a late lunch. If five students have burgers and three have sub sandwiches, the bill will be $35. If three students have burgers and five have sub sandwiches, the bill will be $37. What is the cost of each type sandwich?
Interpret the equation directly again:
Let x = the cost of a burger.
Let y = the cost of a sub sandwich.
5x + 3y = 35
3x + 5y = 37
This was solved using elimination with multiplication.
The solution is (4, 5).
A burger will cost $4 and a sub sandwich will cost $5.
Quick Practice
Set up each of the following problems as a system of equations in two variables, x and y.
The total number of students and adults who were at a concert was 480. The cost of tickets was $15 per student and $24 per adult. The total amount of income from tickets was $9,270. How many students and how many adults were at the concert?
When you have a total, such as the total number at the concert that is usually one of the equations.
Let x = # students.
Let y = # adults.
The other equation is the cost.
The length of a rectangle is 16 centimeters (cm) longer than twice its width. The perimeter is 122 cm. Find the dimensions of the rectangle.
Use the first sentence for one equation, using L and W for length and width.
Using the formula for perimeter of a rectangle:
Ace Cheeses produces grated cheese made from two types of cheese. One type of cheese costs $3.10 per pound (lb.) and another type $2.90 per pound. How many pounds of each should be mixed to obtain 200 pounds worth $2.95 per pound?
Write one equation for the total pounds. Let the number of pounds of the two types of cheese be A and B.
The other equation will come from the value of each and total cost.
An artist has 40 kilograms (kg) of an alloy that is 65% copper. How many kilograms of a second alloy, that is 42% copper, should be mixed with the first alloy to get a new alloy that is 50% copper?
Again, write one equation for the total and another for the value, this time using percents. Let A stand for the number of kg of the 42% alloy and B stand for the number of kg of the final (new) alloy.
The total cost of your car insurance and registration is $580. The cost of the insurance is eight times the cost of registration. What is the cost of each?
Write one equation for total cost; the other will be the relationship in the second sentence. Let x be the cost of the insurance and y the cost of the registration.
For questions 6 through 10, use any technique to solve each of the systems set up in questions 1 through 5.
The total number of students and adults who were at a concert was 480. The cost of tickets was $15 per student and $24 per adult. The total amount of income from tickets was $9,270. How many students and how many adults were at the concert?
One way to solve this might be elimination. Multiply the first equation by 15, and then subtract.
There were 250 students and 230 adults at the concert.
The length of a rectangle is 16 cm longer than twice its width. The perimeter is 122 cm. Find the dimensions of the rectangle.
One way to solve this is to use substitution.
The length is 46 cm and the width is 15 cm.
Ace Cheeses produces grated cheese made from two types of cheese. One type of cheese costs $3.10 per pound and another type $2.90 per pound. How many pounds of each should be mixed to obtain 200 pounds worth $2.95 per pound?
You could use elimination here, multiplying the first equation by 3.10, and then subtracting.
Fifty pounds of the first cheese and 150 pounds of the second cheese will be used in the mixture.
An artist has 40 kg of an alloy that is 65% copper. How many kilograms of a second alloy, that is 42% copper, should be mixed with the first alloy to get a new alloy that is 50% copper?
Using substitution,
Seventy-five kilograms of the 42% alloy should be used, which would make 115kg of the 50% alloy.
The total cost of your car insurance and registration is $580. The cost of the insurance is eight times the cost of registration. What is the cost of each?
Using substitution,
So, rounding the answers to dollars and cents, the cost of insurance was about $515.56 and the registration was about $64.44.
Problem Solving 1 (10 points)
Find the 6-Problem Solving 1 link. You may do this more than one time to improve your score.
Section Homework (10 points)
It’s time to complete your homework. Find the 6-Section Homework link to submit it for a grade. You may take this only one time, so check your understanding of the material before completing your homework, and do your best.
Problem Solving 1 (
Section Homework (