Use the elimination method to find all solutions of the system of equations. { 4 x-3 y=11

Use the elimination method to find all solutions of the...

Key Concepts

Elimination Method

The elimination method is a systematic procedure for solving systems of linear equations by eliminating one variable at a time. This is done by adding or subtracting multiples of equations to cancel out one variable, thereby reducing the system to a simpler form that is easier to solve.

System of Linear Equations

A system of linear equations is a set of equations where each equation is linear in the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously, which typically represents the intersection point(s) of the corresponding lines or planes.

Transcript

00:01 So my objective on this problem is to use the elimination method to find all of the possible solutions to the system of equations.

00:08 So first, let's just review a system of equations is when you have two or more equations in the same variable.

00:14 So here i have my first equation, 4x minus 3 .i equals 11.

00:18 My second equation, 8x plus 4i equals 12.

00:21 And they both use the variables x and y.

00:23 So therefore, i have a system.

00:25 A solution to a system of equations is where these two equations, which both represent, represent lines where those two lines would intersect.

00:34 So it's the point of intersection.

00:36 So ultimately i'm searching for the x comma y of point of intersection.

00:41 And there's three ways that we can solve a system of equations.

00:44 We can graph where we actually graph and see where they intersect.

00:48 We can substitute, but in order to do that, one of our variables would need to be isolated or we can eliminate.

00:53 Now when we eliminate, what we do is we either add these equations together or subtract them from each other in the hopes that one of our variables will be eliminated.

01:03 Currently, if i were to add 4x plus 8x is 12x, that would not eliminate.

01:09 Negative 3y plus 4y would be 1y, that would not eliminate.

01:12 Same with subtracting, 4 minus 8 would be negative 4, and negative 3 minus 4 would be negative 7.

01:19 So right now i'm not ready to eliminate.

01:22 But if i can get my coefficients of either x or y to match in their absolute value, then i'll be able to eliminate by either adding or subtracting.

01:32 So this is kind of like finding a least common multiple or almost like finding a least common denominator if you were adding fractions.

01:40 So i can see that x's coefficients are 4 and 8, which means i can make the coefficients of x both into 8 by changing my top equation.

01:50 Y has a coefficient of negative 3 and 4, and so the smallest number i could do there is 12, but that would mean i would need to change both equations.

01:58 I'm going to choose to change this coefficient of x to an eighth, and in order to do that, i would multiply by 2.

02:06 And so basically what i'm doing here is i'm changing my standard form equation into an equivalent equation by multiplying the entire thing by 2.

02:14 It's important to know that when i do this, i need to make sure i multiply every single component by 2.

02:20 So that top equation is now going to become 8x minus 6y equals 3.

02:29 22.

02:31 I'm just going to create a little bit of room there.

02:33 Now the bottom equation doesn't need to change.

02:35 So i'm just going to bring that over.

02:37 8x plus 4y equals 12.

02:42 Now i'm going to draw my line.

02:44 And something i like to do as a strategy is i like to draw a little circle here...