Let's solve each expression step by step using the order of operations and the values for \(x\), \(y\), and \(z\) given: \(x = 5.3\), \(y = -1\), and \(z = 7\).
1. Calculate \(3x + 4y\)
First, substitute the values for \(x\) and \(y\):
\[ 3x + 4y = 3(5.3) + 4(-1) \]
Calculate \(3(5.3)\):
\[ 3(5.3) = 15.9 \]
Calculate \(4(-1)\):
\[ 4(-1) = -4 \]
Now add the results:
\[ 15.9 + (-4) = 15.9 - 4 = 11.9 \]
So,
\[ 3x + 4y = 11.9 \]
2. Calculate \(xyz\)
Now, substitute \(x\), \(y\), and \(z\):
\[ xyz = 5.3 \times (-1) \times 7 \]
Calculate \(5.3 \times (-1)\):
\[ 5.3 \times (-1) = -5.3 \]
Now calculate \(-5.3 \times 7\):
\[ -5.3 \times 7 = -37.1 \]
So,
\[ xyz = -37.1 \]
3. Calculate \(-\frac{2y}{y - 2z - y^2}\)
Substitute the values for \(y\) and \(z\):
\[ -\frac{2y}{y - 2z - y^2} = -\frac{2(-1)}{-1 - 2(7) - (-1)^2} \]
Calculate the numerator:
\[ -2(-1) = 2 \]
Now calculate the denominator step by step:
- \(y = -1\)
- \(2z = 2(7) = 14\)
- \(y^2 = (-1)^2 = 1\)
Now substitute these values into the denominator:
\[ -1 - 14 - 1 = -1 - 14 - 1 = -16 \]
Now we have:
\[ -\frac{2}{-16} \]
This simplifies to:
\[ \frac{2}{16} = \frac{1}{8} \]
So,
\[ -\frac{2y}{y - 2z - y^2} = \frac{1}{8} \]
Summary of Results:
- \(3x + 4y = 11.9\)
- \(xyz = -37.1\)
- \(-\frac{2y}{y - 2z - y^2} = \frac{1}{8}\)
These expressions have been calculated using the order of operations and the provided values for \(x\), \(y\), and \(z\).