The substitution walk
To name a letter is to write it using the other letter. One equation can do this. Work each of the three moves.
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x − y = 1 rearranges to x = y + 1.x − y = 1rearranges tox = y + 1.- Put
y + 1for x in2x + 3y = 12. 2(y + 1) + 3y = 12solves to y = 2.- Walk back: x = y + 1 = 3.
The rule
Rule: Rearrange one equation to name a letter. Put that name into the other equation.
Putting one name in place of a letter is called substitution.
Take 2x + 3y = 12 and x − y = 1. In x − y = 1, x has coefficient 1. So name it: x = y + 1. Put that name into the other equation: 2(y + 1) + 3y = 12. Open the bracket: 2y + 2 + 3y = 12. So 5y + 2 = 12, then 5y = 10 and y = 2. Walk back: put y = 2 in x = y + 1. That gives x = 3.
| Equation | At (3, 2) | Claim says |
|---|---|---|
2x + 3y = 12 | 6 + 6 = 12 | 12 |
x − y = 1 | 3 − 2 = 1 | 1 |
This method is easiest when a coefficient is 1 or −1. There are no fractions before the substitution starts.
Watch out!
Putting y + 1 back into the same equation gives (y + 1) − y = 1. That is always true and finds nothing new. Put the name into the OTHER equation.