Solve a pair of equations using substitution: word problems
key notes:
To solve using substitution, follow these four steps:
- Step 1: Isolate a variable.
- Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
- Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
- Step 4: State the solution.
Learn with an example
Write simultaneous equations to describe the situation below, solve using substitution, and fill in the blanks.
Zach and his cousin Annie are picking apples in their grandparents’ orchard. Zach has filled 10 baskets with apples and is filling them at a rate of 3 baskets per hour. Annie has 12 full baskets and will continue picking at 1 basket per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch.
How long will that take? How much fruit will they have picked by then?
In ____ hours, the cousins will each have filled _____ baskets with apples.
Before you can solve it, you must write simultaneous equations. Let x represent hours, and let y represent baskets.
y = 3x + 10y = x + 12
Now use substitution to solve the simultaneous equations.
Step 1: Isolate a variable.
The variable y is already isolated in the first equation.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Plug y = 3x + 10 into the other equation, y = x + 12, and find the value of x.
y = x + 12
3x+10 = x + 12 —-> Plugin y = 3x + 10
2x + 10 = 12 ——>Subtract x from both sides
2x = 2 ——> Subtract 10 from both sides
x = 1 ——> Divide both sides by 2
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
Take the result of Step 2, x = 1, and plug it into one of the original equations, such as y = 3x + 10. Then find the value of y.
y = 3x + 10
y = 3(1) + 10 —–> Plug in x = 1
y = 3 + 10 ——->Multiply
y = 13 —–>Add
Step 4: State the solution.
Since x = 1 and y = 13, the solution is (1, 13).
In 1 hour, the cousins will each have filled 13 baskets with apples.
Write simultaneous equations to describe the situation below, solve using substitution, and fill in the blanks.
Jonah and his good buddy Xavier are both mechanics at a shop that does oil changes. They are in a friendly competition to see who can complete the most oil changes in one day. Jonah has already finished 6 oil changes today, and can complete more at a rate of 1 oil change per hour. Xavier just came on shift, and can finish 3 oil changes every hour. Sometime during the day, the friends will be tied, with the same number of oil changes completed. How long will that take?
How many oil changes will Jonah and Xavier each have done?
In _____hours, both men will have completed _____oil changes.
Before you can solve, you must write simultaneous equations. Let x represent hours, and let y represent oil changes.
y = x + 6y = 3x
Now use substitution to solve the simultaneous equations.
Step 1: Isolate a variable.
The variable y is already isolated in the first equation.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Plug y = x + 6 into the other equation, y = 3x, and find the value of x.
y = 3x
x+6 = 3x —–> Plugin y = x + 6
6 = 2x ——> Subtract x from both sides
3 = x ——> Divide both sides by 2
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
Take the result of Step 2, x = 3, and plug it into one of the original equations, such as y = x + 6. Then find the value of y.
y = x + 6
y = 3 + 6 —-> Plug in x = 3
y = 9 —–> Add
Step 4: State the solution.
Since x = 3 and y = 9, the solution is (3, 9).
In 3 hours, both men will have completed 9 oil changes.
Write simultaneous equations to describe the situation below, solve using substitution, and fill in the blanks.
A daycare center in Lowell currently has 1 assistant caregiver and 7 senior caregivers. Since demand is high, the owner is going to be hiring 4 assistant caregivers per month and 1 senior caregiver per month. Her goal is to have a larger staff, including an equal number of assistant caregivers and senior caregivers.
How many of each type will there be? How long will that take?
There will be _____of each type of caregiver in ________ months.
Before you can solve, you must write simultaneous equations. Let x represent months, and let y represent caregivers.
y = 4x + 1
y = x + 7
Now use substitution to solve the simultaneous equations.
Step 1: Isolate a variable.
The variable y is already isolated in the first equation.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Plug y = 4x + 1 into the other equation, y = x + 7, and find the value of x.
y = x + 7
4x + 1 = x + 7 Plug in y = 4x + 1
3x + 1 = 7 Subtract x from both sides
3x = 6
Subtract 1 from both sides
x = 2
Divide both sides by 3
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
Take the result of Step 2, x = 2, and plug it into one of the original equations, such as y = 4x + 1. Then find the value of y.
y = 4x + 1
y = 4(2) + 1 Plug in x = 2
y = 8 + 1 Multiply
y = 9 Add
Step 4: State the solution.
Since x = 2 and y = 9, the solution is (2, 9).
There will be 9 of each type of caregiver in 2 months.
Let’s practice:
