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Math  /  Algebra

QuestionWrite the system of equations as an augmented matrix. {4x−4y=−44−7x−y=−99\left\{\begin{array}{l} 4 x-4 y=-44 \\ -7 x-y=-99 \end{array}\right. □\square □\square Reduce the matrix into reduced row echelon form. □\square □\square □\square □\square Determine the solution to the original system of equations. (x,y)=(x, y)= □\square

Studdy Solution

STEP 1

1. We are given a system of linear equations and need to express it as an augmented matrix.
2. The goal is to convert the matrix to reduced row echelon form (RREF) to find the solution.
3. The solution will be determined from the RREF matrix.

STEP 2

1. Write the system of equations as an augmented matrix.
2. Perform row operations to convert the matrix into reduced row echelon form.
3. Interpret the reduced row echelon form to determine the solution to the system of equations.

STEP 3

Write the given system of equations as an augmented matrix. The system is:
{4x−4y=−44−7x−y=−99\begin{cases} 4x - 4y = -44 \\ -7x - y = -99 \end{cases}
The augmented matrix is:
[4−4∣−44−7−1∣−99]\begin{bmatrix} 4 & -4 & | & -44 \\ -7 & -1 & | & -99 \end{bmatrix}

STEP 4

Perform row operations to convert the matrix into reduced row echelon form (RREF).
First, let's make the leading coefficient of the first row a 1 by dividing the entire first row by 4:
[1−1∣−11−7−1∣−99]\begin{bmatrix} 1 & -1 & | & -11 \\ -7 & -1 & | & -99 \end{bmatrix}

STEP 5

Eliminate the xx-term from the second row by adding 7 times the first row to the second row:
[1−1∣−110−8∣−176]\begin{bmatrix} 1 & -1 & | & -11 \\ 0 & -8 & | & -176 \end{bmatrix}

STEP 6

Make the leading coefficient of the second row a 1 by dividing the entire second row by −8-8:
[1−1∣−1101∣22]\begin{bmatrix} 1 & -1 & | & -11 \\ 0 & 1 & | & 22 \end{bmatrix}

STEP 7

Eliminate the yy-term from the first row by adding the second row to the first row:
[10∣1101∣22]\begin{bmatrix} 1 & 0 & | & 11 \\ 0 & 1 & | & 22 \end{bmatrix}

STEP 8

Interpret the reduced row echelon form to determine the solution to the system of equations. The matrix corresponds to the equations:
{x=11y=22\begin{cases} x = 11 \\ y = 22 \end{cases}
Thus, the solution to the system of equations is:
(x,y)=(11,22)(x, y) = (11, 22)

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