91Ó°ÊÓ

To solve a system of equations using matrices, we often use an "augmented matrix". This is just a fancy name for a matrix that includes both the coefficients of the variables and the constant terms from the equations. Here's how it works:
- List the coefficients of each variable in their respective columns.- Add a vertical line to separate these from the constants on the right.
In the given problem, the system of equations is written as:\[\begin{align*}-x - y &= -10 \3x + 4y &= 24\end{align*}\]The corresponding augmented matrix looks like this:\[\begin{bmatrix}-1 & -1 & | & -10 \3 & 4 & | & 24\end{bmatrix}\]This compact format helps us apply matrix row operations efficiently as we aim to simplify our matrix and eventually solve for the variable values.

A "system of equations" is just a set of two or more equations with the same set of variables. The main goal when working with a system of equations is to find the values of the variables that satisfy all the equations simultaneously.
In the example, our system is made up of two equations but two unknowns:

• \(-x - y = -10\)
• \(3x + 4y = 24\)

These equations describe lines in a coordinate system, and solving the system means finding the point where these two lines intersect. That intersection point will give us the solution where both equations are true at the same time. Generally, you can use various methods to solve them, like graphing, substitution, or matrix row operations, the latter being used in this instance.

Once you've transformed an augmented matrix into a form where one row contains a single variable, "back substitution" allows you to find the solution.
In our example, after performing row operations, we reached the simplified system:

• \(x + \frac{4}{3}y = 8\)
• \(-\frac{1}{3}y = 2\)

Starting with the second equation, solve for \(y\) since it’s isolated:\(-\frac{1}{3}y = 2\)leads to\(y = \frac{-2}{-\frac{1}{3}} = -6\).
Then use the found value of \(y\) in the first equation to solve for \(x\):\[x + \frac{4}{3}(-6) = 8\]Solve for \(x\):\[x = 8 + 8 = 16\]In this step, we substitute backward into the equations to find each variable's value, hence the name "back substitution".