To find the value of the expression [tex]\(\frac{3 x - 5 y^2 - 2 x y z}{\frac{x}{y} - \frac{y^2}{z}}\)[/tex] when [tex]\(x = 3\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(z = 2\)[/tex], we need to evaluate both the numerator and the denominator separately before dividing them.
### Step-by-Step Solution
#### Step 1: Calculate the Numerator
The numerator of the expression is:
[tex]\[ 3 x - 5 y^2 - 2 x y z \][/tex]
Substitute the values [tex]\(x = 3\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(z = 2\)[/tex]:
[tex]\[ 3(3) - 5(-4)^2 - 2(3)(-4)(2) \][/tex]
Let's break it down:
1. Compute [tex]\(3(3)\)[/tex]:
[tex]\[ 3 \times 3 = 9 \][/tex]
2. Compute [tex]\(-5(-4)^2\)[/tex]:
[tex]\[ -4^2 = 16 \][/tex]
[tex]\[ -5 \times 16 = -80 \][/tex]
3. Compute [tex]\(-2(3)(-4)(2)\)[/tex]:
[tex]\[ -2 \times 3 \times -4 = 24 \][/tex]
[tex]\[ 24 \times 2 = 48 \][/tex]
Combining all terms:
[tex]\[ 9 - 80 + 48 = -23 \][/tex]
So, the numerator is [tex]\(-23\)[/tex].
#### Step 2: Calculate the Denominator
The denominator of the expression is:
[tex]\[ \frac{x}{y} - \frac{y^2}{z} \][/tex]
Substitute the values [tex]\(x = 3\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(z = 2\)[/tex]:
[tex]\[ \frac{3}{-4} - \frac{(-4)^2}{2} \][/tex]
Let's break it down:
1. Compute [tex]\(\frac{3}{-4}\)[/tex]:
[tex]\[ \frac{3}{-4} = -0.75 \][/tex]
2. Compute [tex]\( \frac{(-4)^2}{2} \)[/tex]:
[tex]\[ -4^2 = 16 \][/tex]
[tex]\[ \frac{16}{2} = 8 \][/tex]
Combining both terms:
[tex]\[ -0.75 - 8 = -8.75 \][/tex]
So, the denominator is [tex]\(-8.75\)[/tex].
#### Step 3: Divide the Numerator by the Denominator
Finally, we divide the calculated numerator by the calculated denominator:
[tex]\[ \frac{-23}{-8.75} \][/tex]
Dividing these values gives us:
[tex]\[ \frac{-23}{-8.75} \approx 2.6285714285714286 \][/tex]
### Conclusion
The value of the expression [tex]\(\frac{3 x - 5 y^2 - 2 x y z}{\frac{x}{y} - \frac{y^2}{z}}\)[/tex] when [tex]\(x = 3\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(z = 2\)[/tex] is approximately [tex]\(2.6285714285714286\)[/tex].
