x+y=10
y+z=20
x+z=24
x+y+z=
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MathBot Answer:
The system of linear equations has exactly one solution.
The substitutions are \( x = 7 \), \( y = 3 \), \( z = 17 \).
\[x + y + z = 27\]
Begin by writing the augmented matrix of the system of equations. $$\begin{bmatrix}1 & 1 & 0 &\bigm |& 10\\0 & 1 & 1 &\bigm |& 20\\1 & 0 & 1 &\bigm |& 24\end{bmatrix}$$
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.
$$\begin{bmatrix}1 & 1 & 0 &\bigm |& 10\\0 & 1 & 1 &\bigm |& 20\\1 & 0 & 1 &\bigm |& 24\end{bmatrix}$$
The leading term of row \(1\) is already \(1\) so this row does not need to be multiplied by a scalar.
$$\begin{bmatrix}1 & 1 & 0 &\bigm |& 10\\0 & 1 & 1 &\bigm |& 20\\1 & 0 & 1 &\bigm |& 24\end{bmatrix}$$
Multiply row \(1\) by scalar \(-1\) and add it to row \(3\).
$$\begin{bmatrix}1 & 1 & 0 &\bigm |& 10\\0 & 1 & 1 &\bigm |& 20\\0 & -1 & 1 &\bigm |& 14\end{bmatrix}$$
Switch the rows in the matrix such that the row with the next leftmost non-zero entry with the next greatest magnitude is the next row from the top.
$$\begin{bmatrix}1 & 1 & 0 &\bigm |& 10\\0 & 1 & 1 &\bigm |& 20\\0 & -1 & 1 &\bigm |& 14\end{bmatrix}$$
The leading term of row \(2\) is already \(1\) so this row does not need to be multiplied by a scalar.
$$\begin{bmatrix}1 & 1 & 0 &\bigm |& 10\\0 & 1 & 1 &\bigm |& 20\\0 & -1 & 1 &\bigm |& 14\end{bmatrix}$$
Multiply row \(2\) by scalar \(-1\) and add it to row \(1\).
$$\begin{bmatrix}1 & 0 & -1 &\bigm |& -10\\0 & 1 & 1 &\bigm |& 20\\0 & -1 & 1 &\bigm |& 14\end{bmatrix}$$
Multiply row \(2\) by scalar \(1\) and add it to row \(3\).
$$\begin{bmatrix}1 & 0 & -1 &\bigm |& -10\\0 & 1 & 1 &\bigm |& 20\\0 & 0 & 2 &\bigm |& 34\end{bmatrix}$$
Multiply row \(3\) by scalar \(\frac{1}{2}\) to make the leading term \(1\).
$$\begin{bmatrix}1 & 0 & -1 &\bigm |& -10\\0 & 1 & 1 &\bigm |& 20\\0 & 0 & 1 &\bigm |& 17\end{bmatrix}$$
Multiply row \(3\) by scalar \(1\) and add it to row \(1\).
$$\begin{bmatrix}1 & 0 & 0 &\bigm |& 7\\0 & 1 & 1 &\bigm |& 20\\0 & 0 & 1 &\bigm |& 17\end{bmatrix}$$
Multiply row \(3\) by scalar \(-1\) and add it to row \(2\).
$$\begin{bmatrix}1 & 0 & 0 &\bigm |& 7\\0 & 1 & 0 &\bigm |& 3\\0 & 0 & 1 &\bigm |& 17\end{bmatrix}$$
Once the matrix is in reduced-row echelon form, convert the matrix back into linear equations to find the solution. $$\begin{aligned}1 \cdot x+ 0 \cdot y+ 0 \cdot z = 7 \\ x = 7\end{aligned}$$$$\begin{aligned}0 \cdot x+ 1 \cdot y+ 0 \cdot z = 3 \\ y = 3\end{aligned}$$$$\begin{aligned}0 \cdot x+ 0 \cdot y+ 1 \cdot z = 17 \\ z = 17\end{aligned}$$
By substituting these values into the expression, we find that:$$\begin{aligned} x + y + z &= 3 + 7 + 17\\ &=27\end{aligned}$$