Unit 4, Lesson 14
Solving Systems by Elimination (Part 1)
Integrated Math 1
14.1
Warm-up
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
Another way to solve systems of equations algebraically is by elimination. Just like in substitution, the idea is to eliminate one variable so that we can solve for the other. This is done by adding or subtracting equations in the system. Let’s look at an example.
Notice that one equation has and the other has .
If we add the second equation to the first, the and add up to 0, which eliminates the -variable, allowing us to solve for .
Now that we know , we can substitute 10 for in either of the equations and find :
In this system, the coefficient of in the first equation happens to be the opposite of the coefficient of in the second equation. The sum of the terms with -variables is 0.
What if the equations don't have opposite coefficients for the same variable, like in the following system?
Notice that both equations have , and if we subtract the second equation from the first, the variable will be eliminated because is 0.
Substituting 5 for in one of the equations gives us :
Adding or subtracting the equations in a system creates a new equation. How do we know the new equation shares a solution with the original system?
If we graph the original equations in the system and the new equation, we can see that all three lines intersect at the same point, but why do they?
Three intersecting lines on coordinate plane, origin O. X axis from negative 4 to 14 by 2’s. Y axis from negative 4 to 12 by 2’s. Black line through negative 4 comma 12 and 10 comma 2. Blue line passes through negative 4 comma 1 and 10 comma 2. Red vertical line passes through 10 comma 2. All lines intersect at 10 comma 2.
In future lessons, we will investigate why this strategy works.
Elimination is a method of solving a system of two equations in two variables. A multiple of one equation is added to or subtracted from another to get an equation with only one of the variables. (The other variable is eliminated.)