Solve the system of linear equation. any solution algebraically. { 2 x +2 z=2 5 x+3 y

step 1 So we're given this system of equations here. I'm going to take liberty of writing them down. An

Solve the system of linear equation. any solution...

Key Concepts

System of Linear Equations

This refers to a set of equations, all of which share the same set of unknown variables. The goal is to find values for these variables that simultaneously satisfy every equation in the system. These types of systems are fundamental in algebra and can be solved using various methods depending on their structure and the number of equations relative to the unknowns.

Elimination Method

This is a technique used to solve systems of equations by adding or subtracting equations to eliminate one or more of the unknowns. The method involves manipulating the equations so that one variable cancels out, reducing the system to fewer equations which can be solved more easily.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others and then substituting that expression into the remaining equations. This helps to reduce the system step by step until it becomes a single equation in one variable, which can then be solved directly.

Consistent and Unique Solutions

A consistent system is one that has at least one set of solutions. If the system has exactly one solution, it is considered to have a unique solution. Recognizing whether a system of linear equations is consistent and whether it has a unique solution is important in determining the appropriate method for solving it.

Transcript

00:01 So we're given this system of equations here.

00:05 I'm going to take liberty of writing them down.

00:07 And we're asked to solve this algebraically.

00:11 All right, i'm going to call this equation one, equation two, and equation three.

00:17 And what do i notice? well, i notice that equation one does not have a y.

00:22 Two and three both have a y with the same coefficient.

00:26 So if i could subtract one from the other, then that would be gone.

00:32 And the result will just have x and z in it which i could use with equation one to solve so let's do equation two minus equation three so that's five x three y minus three y that's gone zero minus a negative four x four z that's four z and four minus four is zero i want to call this equation four so i want to combine equation 4 and equation 1 together.

01:08 And what do i notice? well, equation 1 has a 2z.

01:12 Equation 4 has a 4z.

01:14 So if i could do minus 2 times equation 1, i could add them together, and the zs would be gone, wouldn't they? all right, so minus 2 times 2x, that's minus 4x, minus 4z by design.

01:31 2 times minus 2, that's minus 4.

01:35 Add these together.

01:37 Of course the x's are gone, right? i'm sorry, not the x's.

01:41 Oops.

01:43 The zs will be gone.

01:44 The zs.

01:45 Z will be gone here...

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