To solve this system of equations, we need to solve for the variables [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] such that all three equations are satisfied. The system of equations is:
[tex]\[
\begin{array}{l}
x + 2y - 6 = z \\
3y - 2z = 7 \\
4 + 3x = 2y - 5z
\end{array}
\][/tex]
Let's compare the given options to the correct solution.
First, we rewrite the equations:
1. [tex]\( x + 2y - z = 6 \)[/tex]
2. [tex]\( 3y - 2z = 7 \)[/tex]
3. [tex]\( 3x - 2y + 5z = -4 \)[/tex]
Substituting the given options into each equation:
### Option A:
[tex]\( x = 0, y = 5, z = 4 \)[/tex]
1. [tex]\( 0 + 2(5) - 4 = 6 \)[/tex]
[tex]\[
2 \cdot 5 - 4 = 10 - 4 = 6 \quad \text{(True)}
\][/tex]
2. [tex]\( 3(5) - 2(4) = 7 \)[/tex]
[tex]\[
3 \cdot 5 - 2 \cdot 4 = 15 - 8 = 7 \quad \text{(True)}
\][/tex]
3. [tex]\( 3(0) - 2(5) + 5(4) = -4 \)[/tex]
[tex]\[
- 2 \cdot 5 + 5 \cdot 4 = -10 + 20 = 10 \quad \text{(False)}
\][/tex]
Since the third equation is false, Option A is not correct.
### Option B:
[tex]\( x = 2, y = 1, z = 2 \)[/tex]
1. [tex]\( 2 + 2(1) - 2 = 6 \)[/tex]
[tex]\[
2 + 2 - 2 = 2 \quad \text{(False)}
\][/tex]
Since the first equation is false, Option B is not correct.
### Option C:
[tex]\( x = \frac{2}{3}, y = \frac{3}{2}, z = -\frac{23}{6} \)[/tex]
1. [tex]\( \frac{2}{3} + 2 \left(\frac{3}{2}\right) - \left(-\frac{23}{6}\right) = 6 \)[/tex]
[tex]\[
\frac{2}{3} + 3 + \frac{23}{6} = \frac{4}{6} + \frac{18}{6} + \frac{23}{6} = \frac{45}{6} = 7.5 \quad \text{(False)}
\][/tex]
Since the first equation is false, Option C is not correct.
### Option D:
[tex]\( x = \frac{7}{4}, y = \frac{3}{2}, z = -\frac{5}{4} \)[/tex]
1. [tex]\( \frac{7}{4} + 2 \left(\frac{3}{2}\right) - \left(-\frac{5}{4}\right) = 6 \)[/tex]
[tex]\[
\frac{7}{4} + 3 + \frac{5}{4} = \frac{7}{4} + \frac{12}{4} + \frac{5}{4} = \frac{24}{4} = 6 \quad \text{(True)}
\][/tex]
2. [tex]\( 3 \left(\frac{3}{2}\right) - 2 \left(-\frac{5}{4}\right) = 7 \)[/tex]
[tex]\[
\frac{9}{2} + \frac{10}{4} = \frac{18}{4} + \frac{10}{4} = \frac{28}{4} = 7 \quad \text{(True)}
\][/tex]
3. [tex]\( 3 \left(\frac{7}{4}\right) - 2 \left(\frac{3}{2}\right) + 5 \left(-\frac{5}{4}\right) = -4 \)[/tex]
[tex]\[
\frac{21}{4} - 3 - \frac{25}{4} = \frac{21}{4} - \frac{12}{4} - \frac{25}{4} = \frac{21 - 12 - 25}{4} = \frac{-16}{4} = -4 \quad \text{(True)}
\][/tex]
Since all equations hold true, Option D [tex]\( \left( x = \frac{7}{4}, y = \frac{3}{2}, z = -\frac{5}{4} \right) \)[/tex] is correct. Therefore, the correct option is:
D. [tex]\( x = \frac{7}{4}, y = \frac{3}{2}, z = -\frac{5}{4} \)[/tex]
