Solve using augmented matrices: -x - 6y = 14 and y = -4.
Understand the Problem
The question is asking to solve a system of linear equations represented in augmented matrix form. Specifically, it involves two equations: -x - 6y = 14 and y = -4, and requires using matrix techniques to find the values of x and y.
Answer
The solution is \(x = 10\) and \(y = -4\).
Answer for screen readers
The solution to the system of equations is: $$ x = 10, ; y = -4 $$
Steps to Solve
- Set Up the Augmented Matrix
The equations given are: $$ -x - 6y = 14 $$ $$ y = -4 $$
We can rewrite the first equation to isolate the terms for the matrix: $$ -x - 6y + 0 = 14 $$ In matrix form, the augmented matrix becomes: $$ \begin{bmatrix} -1 & -6 & | & 14 \ 0 & 1 & | & -4 \end{bmatrix} $$
- Perform Row Operations
Next, we want to use row operations to simplify this matrix. We can start by making the first element of the first row a 1. We do this by multiplying the first row by -1: $$ R_1 \to -1 R_1 $$ This transforms our matrix to: $$ \begin{bmatrix} 1 & 6 & | & -14 \ 0 & 1 & | & -4 \end{bmatrix} $$
- Eliminate y from Row 1
Now, we need to eliminate the second variable from the first row. We can do this by subtracting 6 times the second row from the first row: $$ R_1 \to R_1 - 6R_2 $$ This gives us: $$ \begin{bmatrix} 1 & 0 & | & -14 + 24 \ 0 & 1 & | & -4 \end{bmatrix} $$ Which simplifies to: $$ \begin{bmatrix} 1 & 0 & | & 10 \ 0 & 1 & | & -4 \end{bmatrix} $$
- Read the Solution
Now, we can read the solution directly from the augmented matrix:
From the first row: $x = 10$
From the second row: $y = -4$
The solution to the system of equations is: $$ x = 10, ; y = -4 $$
More Information
Using augmented matrices is a great technique for solving systems of linear equations. In this case, we found that (x) equals 10 and (y) equals -4 through systematic row operations.
Tips
- Forgetting to apply row operations correctly: Always ensure that each step transforms the matrix appropriately.
- Misreading the final matrix: Make sure to interpret the values of (x) and (y) correctly from the final augmented matrix.
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