Swap it in
Write one letter in terms of the other, then put it in.
When a graph is not enough
A graph is hard to read when the meeting point has coordinates like (413, 119). We need an exact method. From x + 2y = 3 we can write x = 3 − 2y, so wherever x appears we may write 3 − 2y.
What happens if we do that in 7x − 15y = 2?
What this lesson covers
The idea
Express one variable in terms of the other from either equation, substitute it in the other equation to get an equation in one variable and solve it, then substitute back to find the other variable.
When a graph is not enough
A graph is hard to read when the meeting point has coordinates like (4/13, 1/19). We need an exact method. From x + 2y = 3 we can write x = 3 − 2y, so wherever x appears we may write 3 − 2y.
What happens if we do that in 7x − 15y = 2?
- It becomes an equation with only y
- It still has x and y
- It disappears
Swap it in
Solve each pair. Choose a letter to write in terms of the other, pick the equation to put it into, work it out, then put the value back. Try the other choices too: the answer never changes.
Substitution, step by step
Step 1: write one letter in terms of the other, from either equation. Step 2: substitute it in the other equation. You get an equation in one letter. Solve it. Step 3: put that value back to find the other letter.
The book does 7x − 15y = 2 and x + 2y = 3 like this: x = 3 − 2y 7(3 − 2y) − 15y = 2 21 − 14y − 15y = 2 −29y = −19, so y = 19/29 x = 3 − 2(19/29) = 49/29
Always check by putting x and y into both original equations.
Notes
Step 1: write one letter in terms of the other. Step 2: substitute it in the other equation and solve. Step 3: put the value back to find the other letter.
Check yourself
You have y = 2x + 1 and 3x + y = 11. Which first move is the smooth one?
Put y = 2x + 1 into 3x + y = 11 and solve. What is x?
Answer: 2
3x + 2x + 1 = 11, so 5x = 10 and x = 2.
Now find y from y = 2x + 1.
Answer: 5
y = 2 × 2 + 1 = 5. Check in 3x + y = 11: 6 + 5 = 11 ✓.
Aftab’s age s and his daughter’s age t satisfy s − 7t + 42 = 0 and s − 3t = 6. From the second, s = 3t + 6. Put it into the first. What is t?
Answer: 12
−4t + 48 = 0, so t = 12. Then s = 3 × 12 + 6 = 42: Aftab is 42 and his daughter 12 (the book’s Example 5).
- Put 2x + 1 in place of y in 3x + y = 11 — correct. Yes! That leaves an equation with only x.
- Put 2x + 1 in place of y in y = 2x + 1. That is the equation it came from. It only gives 2x + 1 = 2x + 1, which tells us nothing new.
- Put 11 in place of x. We do not know x yet, and 11 is not x.