To solve the given system of equations, we first rewrite them clearly:
- Equation 1: \(-30 - 4y = -23\)
- Equation 2: \(3z - 5y = 5\)
Let's start with the first equation and solve for \(y\):
Step 1: Solve the first equation for \(y\)
\[ -30 - 4y = -23 \]
Add 30 to both sides:
\[ -4y = -23 + 30 \]
Simplifying gives:
\[ -4y = 7 \]
Now, divide by -4:
\[ y = -\frac{7}{4} \]
Step 2: Substitute \(y\) into the second equation
Now we substitute \(y = -\frac{7}{4}\) into the second equation:
\[ 3z - 5y = 5 \]
Substituting for \(y\):
\[ 3z - 5\left(-\frac{7}{4}\right) = 5 \]
This simplifies to:
\[ 3z + \frac{35}{4} = 5 \]
Step 3: Solve for \(z\)
To get rid of the fraction, multiply the whole equation by 4:
\[ 4(3z) + 35 = 20 \]
This simplifies to:
\[ 12z + 35 = 20 \]
Now, subtract 35 from both sides:
\[ 12z = 20 - 35 \]
This simplifies to:
\[ 12z = -15 \]
Now, divide both sides by 12:
\[ z = -\frac{15}{12} = -\frac{5}{4} \]
Summary of Solutions
We've solved the system of equations:
- \(y = -\frac{7}{4}\)
- \(z = -\frac{5}{4}\)
Thus, the solution to the system of equations is:
\[ \boxed{\left( y = -\frac{7}{4}, z = -\frac{5}{4} \right)} \]