Solve each system of equations. If the system has no solution, say that it is inconsistent. {

Solve each system of equations. If the system has no...

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of equations by adding or subtracting equations to cancel out one or more variables. This reduces the system step-by-step into a simpler form, often a single equation in one variable, which can then be solved directly. This method is particularly useful for systems with multiple equations and variables, allowing for a systematic reduction to find the solution.

Systems of Linear Equations

A system of linear equations consists of two or more equations that involve the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously, representing the point or points where the corresponding graphs intersect. Such systems can be expressed in various forms, including the standard form and matrix form, and they are foundational in algebra and linear algebra.

Consistency and Inconsistency

In the context of systems of equations, a system is considered consistent if it has at least one solution, meaning there exists an assignment of values to the variables that satisfies all equations. Conversely, a system is inconsistent if no such solution exists, meaning the equations contradict each other. Recognizing consistency or inconsistency helps in understanding whether a system can be solved.

Transcript

00:01 In order to solve this system of equations, i will be using a combination of substitution and the elimination method.

00:10 I'm first going to start with using substitution.

00:13 So i'm going to rearrange my first equation in order for it to say y equals.

00:22 And i can do that by simply subtracting 2x from both sides.

00:28 So this will read as y equals negative 2x and then minus 4.

00:38 So all i did there was rearrange.

00:41 Next i am going to substitute.

00:44 Everything that y is equal to where y is located in the next equation.

00:50 So let's write this out.

00:52 I'll have negative 2 times negative 2x minus 4, all in parentheses.

01:02 Plus 4 z equals 0.

01:08 Next, i'll use distributive property.

01:11 So that gives me 4x plus 8 plus 4z equals 0.

01:19 And then i'll subtract 8 from both sides, which leaves me with 4x plus 4z equals negative 8.

01:30 Okay, so now that's my first and my second equation combined.

01:35 Now let's take a look at the third equation, which is 3x minus 2z equals negative 11.

01:44 So these two equations right here, i am going to use the elimination method.

01:53 Now, in order to use the elimination method, i need opposite coefficients.

01:57 And if you look at the z's, i already have a positive and a negative.

02:02 So i'm going to work with that, and i'm going to multiply my second equation.

02:07 Equation by two.

02:10 So let's rewrite this off to the side.

02:13 My first equation will stay the same...

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