Math 1 · Unit 6 · Lesson 3 · part 2 of 5
The idea, and why adding is allowed
The second exact method is elimination , also called linear combination.
The idea, and why adding is allowed
The second exact method is elimination, also called linear combination. Line the two equations up and add them so that one variable cancels out. What is left is one equation in one variable, a Unit 1 problem.
2x + 3y = 12 and 4x − 3y = 6
Look at the y terms: 3y and −3y. They are opposite coefficients, additive inverses (the opposites, from Unit 1, Lesson 1), so they add to 0. Add the two equations straight down: 6x + 0y = 18, so 6x = 18, so x = 3. Back-substitute into either original: 2(3) + 3y = 12, so 3y = 6, so y = 2. Solution (3, 2). Check in both: 6 + 6 = 12, true; 12 − 6 = 6, true.
Why adding is allowedAdding equal things to equal things keeps things equal. The left side of equation 1 equals its right side, and the same for equation 2. So left plus left equals right plus right. Adding two true equations gives another true equation, and at the solution point all three hold.
Make a zero appear. That is the same idea as Unit 1, Lesson 1: add the opposite and the additive identity 0 shows up. Here the opposite is a whole term, 3y and −3y, and when they add to 0 the variable disappears. Two variables become one.