Solving Systems by Substitution and Elimination
You can solve a system without a graph. There are two ways. Both turn two equations into one.
What you'll be able to do
- Solve a system by putting one part into the other.
- Solve a system by adding or taking away.
- Multiply first when no letter drops out.
- Spot when there is no answer, or when every point works.
- Check the pair in both.
The big idea
Swap in. Say one says y = x + 2. Put x + 2 in place of y in the other. Now there is only x. This is called substitution.
An example. Take y = x + 2 and 2x + y = 11. Swap in x + 2: 2x + x + 2 = 11. So 3x = 9 and x = 3. Then y = 3 + 2 = 5.
Add them. Add or take away the two so one letter drops out. In x + 2y = 8 and 3x − 2y = 8, add them: 4x = 16. This is called elimination.
Multiply first. Sometimes no letter drops out yet. Then multiply every term of one of them by the same number.
Finish the job. Find both x and y. Then check the pair in both.
Odd cases. Both letters can drop out. If what is left is false, like 6 = 10, there is no answer. If it is true, like 4 = 4, every point on the line works.
Words you need
- System of equations
- Two equations that must both be true at once.
- Substitution
- Putting what one letter equals into the other one.
- Elimination
- Adding or taking away the two so one letter drops out.
- Solution
- The pair (x, y) that makes both true.
Which way is quicker?
| The system looks like | Try |
|---|---|
| One already says y = ... or x = ... | Substitution |
| The same letter has opposite numbers, like 2y and −2y | Add them |
| The same letter has the same number, like 5x and 5x | Take one away from the other |
| Nothing matches yet | Multiply one first, then add or take away |
Worked examples
Example 1
Solve y = x + 2 and 2x + y = 11.
- Put x + 2 in for y: 2x + x + 2 = 11.
- So 3x = 9 and x = 3.
- Then y = 3 + 2 = 5. The pair is (3, 5).
Answer: x = 3, y = 5
Example 2
Solve x + 2y = 8 and 3x − 2y = 8.
- Add the two. The y terms drop out: 4x = 16.
- So x = 4.
- Then 4 + 2y = 8, so y = 2. The pair is (4, 2).
Answer: x = 4, y = 2
Example 3
Solve 2x + y = 7 and 3x + 2y = 12.
- Multiply the first by 2: 4x + 2y = 14.
- Take away the second: x = 2.
- Then 2(2) + y = 7, so y = 3.
Answer: x = 2, y = 3
Common mistakes
Leaving out parentheses when you swap in. Put y = 2x − 3 into 3x + 2y as 3x + 2(2x − 3).
Finding x and stopping. The answer is a pair. Find y too.
Multiplying only one term. Multiply every term, on both sides.
Reading 0 = 0 as x = 0, y = 0. A true line like 0 = 0 means every point on the line works.
Practice: 20 questions
5 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Solve y = 2x − 3 and 3x + 2y = 22.
- Solve 3x + 2y = 7 and 5x − 2y = 9.
- A café sold x small and y large coffees. It knows y = 40 − x and 2x + 3y = 95. How many of each did it sell? Word problem
- Omar adds x + y = 10 and x − y = 4. He gets 2x = 14 and writes the answer x = 7. What is missing?
- A rope is cut into two pieces, x m and y m long. The pieces fit x = 2y and x + y = 18. How long is each piece? Word problem
Tier 2 Intermediate One more step to think through
- Solve 2x + 3y = 12 and 4x − y = 10.
- Solve 3x + 4y = 10 and 2x − 3y = −16.
- A school buys x boxes of pens and y boxes of pencils. It knows 3x + 2y = 31 and 5x + 4y = 55. How many boxes of each? Word problem
- Hana subtracts 5x + 2y = 16 from 5x − 3y = 1. She writes −y = −15. Find her mistake.
- Solve y = 3x − 2 and 6x − 2y = 4.
Tier 3 Advanced Several steps - find the hidden structure
- Solve 2x − 4y = 6 and x − 2y = 5.
- Solve x/3 + y/2 = 4 and x + y = 10.
- Lina buys x kg of rice and y kg of beans. She knows 1.5x + 2.25y = 15 and x + y = 8. How much of each? Word problem
- The system ax + 2y = 10 and 3x − y = 5 has a solution with x = 2. Find y, then a.
- Ivan multiplies x/2 + y/3 = 4 by 6. He writes 3x + 2y = 4. With x − y = 3, he gets x = 2 and y = −1. Find his mistake.
Tier 4 Challenge Stretch problems, some above grade level
- Solve 37x + 13y = 124 and 13x + 37y = 76.
- Look at the system 2x + ky = c and 4x + 6y = 12. Which k and c give no solution?
- 4 muffins and 3 scones cost $23: 4x + 3y = 23. 3 muffins and 2 scones cost $16.50: 3x + 2y = 16.50. Without finding each price, what do 1 muffin and 1 scone cost together? Word problem
- System A: y = 4x − 7 and 2x + 3y = 21. System B: 5x + 2y = 1 and 3x − 2y = 15. Which has the larger x?
- Solve x + y = 9, y + z = 12 and x + z = 11. Above grade level
Answers
Tier 1 Foundations
- x = 4, y = 5. Put 2x − 3 in for y, with parentheses: 3x + 2(2x − 3) = 22. So 7x − 6 = 22, x = 4 and y = 5.
- x = 2, y = 0.5. (2, 0.5) makes both equations true: check it in 3x + 2y = 7 and 5x − 2y = 9.
- 25 small and 15 large. (25, 15) makes both equations true: check it in y = 40 − x and 2x + 3y = 95.
- x = 7, y = 3. Omar stopped at x. Put x = 7 back in: 7 + y = 10, so y = 3. The solution is (7, 3). (7, 3) makes both equations true: check it in x + y = 10 and x − y = 4.
- 12 m and 6 m. (12, 6) makes both equations true: check it in x = 2y and x + y = 18.
Tier 2 Intermediate
- x = 3, y = 2. Multiply every term of 4x − y = 10 by 3: 12x − 3y = 30. Add it to 2x + 3y = 12: 14x = 42. So x = 3 and y = 2.
- x = −2, y = 4. (−2, 4) makes both equations true: check it in 3x + 4y = 10 and 2x − 3y = −16.
- 7 boxes of pens and 5 boxes of pencils. (7, 5) makes both equations true: check it in 3x + 2y = 31 and 5x + 4y = 55.
- x = 2, y = 3. Hana didn't change the sign of 2y. −3y − 2y = −5y, so −5y = −15. Then y = 3 and x = 2. (2, 3) makes both equations true: check it in 5x − 3y = 1 and 5x + 2y = 16.
- Infinitely many solutions. Put 3x − 2 in for y: 6x − 2(3x − 2) = 4. That gives 4 = 4, which is always true. Both equations are the same line.
Tier 3 Advanced
- No solution. Multiply x − 2y = 5 by 2: 2x − 4y = 10. But the first equation says 2x − 4y = 6. 6 = 10 is false, so there is no solution.
- x = 6, y = 4. (6, 4) makes both equations true: check it in x/3 + y/2 = 4 and x + y = 10.
- 4 kg of rice and 4 kg of beans. (4, 4) makes both equations true: check it in 1.5x + 2.25y = 15 and x + y = 8.
- y = 1 and a = 4. Put x = 2 into 3x − y = 5: 6 − y = 5, so y = 1. Then 2a + 2 = 10, so a = 4.
- x = 6, y = 3. Ivan didn't multiply the 4 by 6. It should be 3x + 2y = 24. That gives x = 6 and y = 3. (6, 3) makes both equations true: check it in x/2 + y/3 = 4 and x − y = 3.
Tier 4 Challenge
- x = 3, y = 1. Add: 50x + 50y = 200, so x + y = 4. Subtract: 24x − 24y = 48, so x − y = 2. Then x = 3 and y = 1.
- k = 3 and any c except 6 give no solution. Double 2x + 3y and you get 4x + 6y. So k = 3 makes the x and y parts match. If c = 6 it is the same line. Any other c makes the lines parallel.
- $6.50. Subtract the second equation from the first: x + y = 23 − 16.50 = 6.50. So one of each costs $6.50.
- System A (x = 3, against x = 2). System A: substitute to get 14x − 21 = 21, so x = 3. System B: add to get 8x = 16, so x = 2. 3 is larger.
- x = 4, y = 5, z = 7. Add all three: 2x + 2y + 2z = 32, so x + y + z = 16. Take away each equation: z = 7, x = 4, y = 5.
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