Solve a pair of equations using elimination: word problems
key notes:
To solve using elimination, follow these four steps:
- Step 1: Make sure the equations have opposite x terms or opposite y terms.
- Step 2: Add to eliminate one variable and solve for the other.
- Step 3: Plug the result of Step 2 into one of the original equations and solve.
- Step 4: State the solution.
Learn with an example
Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks.
An administrative assistant is making some copies. She made 23 one-sided copies and 48 two-sided copies for the V.P. of Marketing, which took a total of 286 seconds. Next, she made 13 one-sided copies and 24 two-sided copies for the Director of Sales, which took 146 seconds.
How long does it take to make each type of copy?
It takes _____seconds to make a one-sided copy and _____seconds to make a two-sided copy.
Before you can solve, you must write simultaneous equations. Let x represent the amount of time to make a one-sided copy, and let y represent the amount of time to make a two-sided copy.
- 23x + 48y = 286
- 13x + 24y = 146
Now use elimination to solve the simultaneous equations.
Step 1: Make sure the equations have opposite x terms or opposite y terms.
Currently, neither the x terms (23x and 13x) nor the y terms (48y and 24y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the second equation by -2.
| 23x + 48y = 286 | → | 23x + 48y = 286 |
| -2(13x + 24y = 146) | → | -26x − 48y = -292 |
Now the y terms (48y and -48y) are opposites.
Step 2: Add to eliminate one variable and solve for the other.
Add to eliminate the y terms, and then solve for x.
23x + 48y = 286
+ ( –26x − 48y = –292 )
–3x + 0y = –6 ——-> Add to eliminate the y terms
–3x = –6 Simplify
x = 2 Divide both sides by –3
Step 3: Plug the result of Step 2 into one of the original equations and solve.
Take the result of Step 2, x = 2, and plug it into one of the original equations, such as 23x + 48y = 286. Then find the value of y.
23x + 48y = 286
23(2) + 48y = 286 —>Plug in x = 2
46 + 48y = 286 —->Multiply
48y = 240 —->Subtract 46 from both sides
y = 5 —–>Divide both sides by 48
Step 4: State the solution.
Since x = 2 and y = 5, the solution is (2, 5).
It takes 2 seconds to make a one-sided copy and 5 seconds to make a two-sided copy.
Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks.
Shannon is selling her handmade jewellery online. Yesterday, she sold 10 bracelets and 4 necklaces, for a profit of ₹188. Today, she made a profit of ₹74 by selling 3 bracelets and 2 necklaces.
How much profit does Shannon earn from each piece?
Shannon earns a profit of ₹_____ from every bracelet and ₹____ from every necklace.
Before you can solve, you must write simultaneous equations. Let x represent the profit on every bracelet, and let y represent the profit on every necklace.
- 10x + 4y = 188
- 3x + 2y = 74
Now use elimination to solve the simultaneous equations.
Step 1: Make sure the equations have opposite x terms or opposite y terms.
Currently, neither the x terms (10x and 3x) nor the y terms (4y and 2y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the second equation by –2.
| 10x + 4y = 188 | → | 10x + 4y = 188 |
| -2(3x + 2y = 74) | → | -6x − 4y = -148 |
Now the y terms (4y and -4y) are opposites.
Step 2: Add to eliminate one variable and solve for the other.
Add to eliminate the y terms, and then solve for x.
10x + 4y = 188
+ ( –6x − 4y = –148 )
4x + 0y = 40 —> Add to eliminate the y terms
4x = 40 ——->Simplify
x = 10 ——-> Divide both sides by 4
Step 3: Plug the result of Step 2 into one of the original equations and solve.
Take the result of Step 2, x = 10, and plug it into one of the original equations, such as 10x + 4y = 188. Then find the value of y.
10x + 4y = 188
10(10) + 4y = 188 —> Plug in x = 10
100 + 4y = 188 —-> Multiply
4y = 88 —-> Subtract 100 from both sides
y = 22 —-> Divide both sides by 4
Step 4: State the solution.
Since x = 10 and y = 22, the solution is (10, 22).
Shannon earns a profit of ₹10 from every bracelet and ₹22 from every necklace.
Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks.
Tina and Elise decided to shoot arrows at a simple target with a large outer ring and a smaller bull’s-eye. Tina went first and landed 4 arrows in the outer ring and 3 arrows in the bull’s-eye, for a total of 307 points. Elise went second and got 1 arrow in the outer ring and 3 arrows in the bull’s-eye, earning a total of 241 points.
How many points is each region of the target worth?
The outer ring is worth _______ points, and the bull’s-eye is worth ___ points.
Before you can solve, you must write simultaneous equations. Let x represent the points awarded for an arrow in the outer ring, and let y represent the points awarded for an arrow in the bulls-eye.
4x + 3y = 307
x + 3y = 241
Now use elimination to solve the simultaneous equations.
Step 1: Make sure the equations have opposite x terms or opposite y terms.
Currently, neither the x terms (4x and x) nor the y terms (3y and 3y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the first equation by –1.
–(4x + 3y = 307)→ –4x − 3y = –307
x + 3y = 241→ x + 3y = 241
Now the y terms (–3y and 3y) are opposites.
Step 2: Add to eliminate one variable and solve for the other.
Add to eliminate the y terms, and then solve for x.
–4x − 3y = –307
+( x + 3y = 241 )
–3x + 0y = –66 Add to eliminate the y terms
–3x = –66 Simplify
x = 22 Divide both sides by –3
Step 3: Plug the result of Step 2 into one of the original equations and solve.
Take the result of Step 2, x = 22, and plug it into one of the original equations, such as 4x + 3y = 307. Then find the value of y.
4x + 3y = 307
4(22) + 3y = 307 Plug in x = 22
88 + 3y = 307 Multiply
3y = 219 Subtract 88 from both sides
y = 73 Divide both sides by 3
Step 4: State the solution.
Since x = 22 and y = 73, the solution is (22, 73).
The outer ring is worth 22 points, and the bull’s-eye is worth 73 points.
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