Solve using augmented matrices. -x - 6y = 14, y = -4
Understand the Problem
The question is asking to solve a system of equations using augmented matrices, specifically given the equations -x - 6y = 14 and y = -4. The high-level approach will involve setting up the augmented matrix for the system and using row operations to solve for the variables.
Answer
The solution is \( x = 10 \) and \( y = -4 \).
Answer for screen readers
The solution to the system of equations is:
$x = 10$ and $y = -4$.
Steps to Solve
-
Set Up the Augmented Matrix
Convert the equations into an augmented matrix format. The equations are:
[ -x - 6y = 14 ] [ y = -4 ]
For the first equation, rewrite it as:
[ -1x - 6y = 14 ]
For the second equation, we can express it as:
[ 0x + 1y = -4 ]
Thus, the augmented matrix is:
$$ \begin{bmatrix} -1 & -6 & | & 14 \ 0 & 1 & | & -4 \end{bmatrix} $$ -
Perform Row Operations
We will perform row operations to simplify the augmented matrix. Start with the matrix:
$$ \begin{bmatrix} -1 & -6 & | & 14 \ 0 & 1 & | & -4 \end{bmatrix} $$
To eliminate the variable (y) from the first row, we can replace Row 1 with (R_1 + 6R_2):
$$ R_1 \leftarrow R_1 + 6R_2 $$
Calculating this gives:
$$ \begin{bmatrix} -1 & 0 & | & -10 \ 0 & 1 & | & -4 \end{bmatrix} $$ -
Solve for Variables
Now we have the augmented matrix in row-echelon form. We can interpret this as:
[ -x = -10 \implies x = 10 ]
And from the second row:
[ y = -4 ]
The solution to the system of equations is:
$x = 10$ and $y = -4$.
More Information
The method of augmented matrices is a powerful tool for solving systems of equations, particularly when there are multiple variables. It simplifies the process by using row operations.
Tips
- Forgetting to convert the equations appropriately into the augmented matrix format.
- Not correctly applying row operations, leading to incorrect results. Remember to carefully perform calculations to avoid mistakes.
AI-generated content may contain errors. Please verify critical information
