Math 1 · Unit 6 · Lesson 2 · part 2 of 4
The idea, and why it is allowed
Graphing shows the answer.
The idea, and why it is allowed
Graphing shows the answer. Substitution finds it exactly. The idea: if one equation tells you what y equals, you can put that expression in place of y in the other equation. Then the other equation has only x in it, and you solve it the Unit 1 way.
y = 2x + 1 and 3x + y = 11
The first equation says y is 2x + 1. In the second, wherever it says y, write 2x + 1 instead: 3x + (2x + 1) = 11. One variable. Solve it: 5x + 1 = 11, so 5x = 10, so x = 2. Then back-substitute: y = 2(2) + 1 = 5. The solution is (2, 5). Check in both: 5 = 2(2) + 1, true. 3(2) + 5 = 11, true.
Why you are allowed to swap it inThe solution point satisfies both equations at once. So whatever y equals in the first equation, it must equal the very same thing in the second, because it is the same y at the same point. Replacing y with 2x + 1 in the second equation does not change what the point is; it just writes the second equation using only x.
Two variables became one. That is the whole trick, and it is the same trick elimination will use in Lesson 3. Once there is one variable, it is a Unit 1 equation.