Solve Problems 5-26 using elimination by addition. x-3 y+z= 4 -x+4 y-4 z= 1

Solve Problems 5-26 using elimination by addition. x-3...

Key Concepts

Systems of Linear Equations

Systems of linear equations consist of multiple equations involving the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously. This forms the basis for modeling real-world problems where several conditions hold at once.

Elimination Method

The elimination method is a systematic approach used to solve systems of linear equations by eliminating one variable at a time. This is achieved by adding or subtracting equations to cancel out a variable, thereby reducing the system to a simpler form until the solution can be easily determined.

Addition Strategy in Elimination

Elimination by addition is a specific strategy where equations are added (or subtracted) directly to cancel out one of the variables. This technique involves aligning coefficients so that at least one variable has equal and opposite coefficients in different equations, making its elimination straightforward.

Coefficient Manipulation

Coefficient manipulation involves multiplying entire equations by appropriate constants to enable the cancellation of a variable when equations are added or subtracted. This is a crucial step in the elimination method to ensure that the chosen variable’s coefficients match in magnitude while differing in sign.

Transcript

00:02 So we've got a system of three equations with three unknowns, and we want to use the method of elimination to solve this.

00:11 So we'll start by adding equation a to equation b to eliminate x.

00:18 So a plus b.

00:22 Let's see.

00:22 We'll go ahead and rewrite them so we can see exactly what we're doing here.

00:25 So a is x minus 3y plus z equals 4.

00:33 And b is negative x plus 4y minus 4 z equals 1.

00:42 So if we add these two together, x minus x is 0, negative 3 plus 4 is positive 1, y, and negative 4 and positive 1 z makes negative 3z equals 4 plus 1 is 5.

00:59 So y minus 3z equals 5, and we'll call that equation d for now, we can also eliminate the x if we use equations b and c.

01:11 And we'll do that by multiplying equation b by 2.

01:16 So 2 times equation b is 2 times negative x is negative 2x.

01:21 And 2 times positive 4y is positive 8y.

01:26 And 2 times negative 4z is negative 8 z.

01:31 And that equals 2 times 1 is 2.

01:33 And we'll go ahead and take c and add.

01:36 That to it.

01:37 So c is positive 2x minus y plus 5 z and that equals negative 3.

01:49 So when we add these two together, positive 8 minus 1 is positive 7 .y.

01:56 Negative 8 and positive 3 is negative 3 z equals 2 plus negative 3 is negative 1.

02:04 So we have 7y minus 3 z equals 7y minus 3 z equals negative 1.