if 23+28+37+x+53=168 and 23+28+40+y+50=120. Find the value of x-y
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MathBot Answer:
The system of linear equations has exactly one solution.
The substitutions are \( x = 27 \), \( y = -21 \).
\[x - y = 48\]
Begin by writing the augmented matrix of the system of equations. $$\begin{bmatrix}1 & 0 &\bigm |& 27\\0 & 1 &\bigm |& -21\end{bmatrix}$$
Then use a series of elementary row operations to convert the matrix into reduced-row echelon form. The three elementary row operations are:
1. Swap the positions of any two rows.
2. Multiply any row by a nonzero scalar.
3. Multiply a row by a nonzero scalar and add it to any other row.
First, switch the rows in the matrix such that the row with the leftmost non-zero entry with the greatest magnitude is at the top.
$$\begin{bmatrix}1 & 0 &\bigm |& 27\\0 & 1 &\bigm |& -21\end{bmatrix}$$
The leading term of row \(1\) is already \(1\) so this row does not need to be multiplied by a scalar.
$$\begin{bmatrix}1 & 0 &\bigm |& 27\\0 & 1 &\bigm |& -21\end{bmatrix}$$
The leading term of row \(2\) is already \(1\) so this row does not need to be multiplied by a scalar.
$$\begin{bmatrix}1 & 0 &\bigm |& 27\\0 & 1 &\bigm |& -21\end{bmatrix}$$
Once the matrix is in reduced-row echelon form, convert the matrix back into linear equations to find the solution. $$\begin{aligned}1 \cdot x+ 0 \cdot y = 27 \\ x = 27\end{aligned}$$$$\begin{aligned}0 \cdot x+ 1 \cdot y = -21 \\ y = -21\end{aligned}$$
By substituting these values into the expression, we find that:$$\begin{aligned} x - y &= 27 - -21\\ &=48\end{aligned}$$