Solve each system analytically. If the equations are...

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable $z$.
$$\begin{aligned}
2 x+3 y+4 z &=3 \\
6 x+3 y+8 z &=6 \\
6 y-4 z &=1
\end{aligned}$$

Key Concepts

Dependent Equations and Parameterization

When the equations in a system are dependent, they do not provide independent information, which leads to infinitely many solutions. In such cases, the solution set is expressed in terms of one or more free parameters, representing the degrees of freedom in the solutions.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equations. This reduces the number of equations and variables, enabling the solver to find the values of the unknowns step by step.

Elimination Method

The elimination method involves combining equations in a system to cancel out one or more variables, simplifying the equations to a form that is easier to solve. By strategically adding or subtracting equations, the number of variables is reduced, facilitating the derivation of solutions.

Analytical Solution

An analytical solution involves solving the system using algebraic methods such as substitution or elimination rather than numerical approximations. This approach provides exact answers structured in a step-by-step logical process.

System of Linear Equations

A system of linear equations involves multiple linear relationships among variables that must be satisfied simultaneously. These systems can be solved using various methods to find unique, infinite, or no solutions, providing insight into the relationships and intersections of planes or lines in space.

Transcript

00:01 Okay, so given these three equations, i notice that if i just add one and three together, i'll cancel z.

00:09 So that's what i'm going to do.

00:10 I'm just going to add equation one plus equation three.

00:14 That'll give me 2x plus there's no x, so we can assume that's 0x.

00:20 So that'll be 2x.

00:22 3y plus 6y is 9y.

00:25 The z's canceled, and this will equal 4.

00:28 Another thing that i'm going to do, since i canceled z from this first equation, i'm also going to try to cancel z from the second equation.

00:37 So i'm going to take the second equation.

00:39 So i'm just going to copy that down.

00:41 6x plus 3y plus 8, z equals 6.

00:47 And then in order to cancel z, i'm going to multiply the third equation by 2 and add them together.

00:55 So 2 times the third equation, there is no x value.

00:58 So it's 0x plus 12y minus 8 z equals 2.

01:06 So now adding these together, 6x plus 0x is 6x plus 15y.

01:15 Xz is cancel, 6 plus 2 is 8.

01:18 I also know that i could divide the left and the right side by 2, so i'm going to divide each term by 2.

01:24 So this will be 3x.

01:27 Oh, sorry, i actually can divide by 2.

01:30 My bad.

01:32 Okay, so now that i have these two equations, what i could do in order to cancel one of them is i could multiply this equation by negative three to try to cancel the x values...

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