Answer :
To solve the given system of equations, we need to find the values of \( x \) and \( y \) that satisfy both equations simultaneously. The system of equations is:
[tex]\[ \begin{array}{r} 2x = 5y + 4 \\ 3x - 2y = -16 \end{array} \][/tex]
Let's follow a step-by-step process to solve this system.
### Step 1: Express one variable in terms of the other from the first equation.
From the first equation:
[tex]\[ 2x = 5y + 4 \][/tex]
We can solve for \(x\):
[tex]\[ x = \frac{5y + 4}{2} \][/tex]
### Step 2: Substitute this expression into the second equation.
Substitute \(x = \frac{5y + 4}{2}\) into the second equation, \(3x - 2y = -16\):
[tex]\[ 3 \left( \frac{5y + 4}{2} \right) - 2y = -16 \][/tex]
### Step 3: Simplify the equation.
Multiply through by 2 to clear the fraction:
[tex]\[ 3(5y + 4) - 4y = -32 \][/tex]
[tex]\[ 15y + 12 - 4y = -32 \][/tex]
[tex]\[ 11y + 12 = -32 \][/tex]
### Step 4: Solve for \(y\).
Isolate \(y\) by subtracting 12 from both sides:
[tex]\[ 11y = -44 \][/tex]
Divide by 11:
[tex]\[ y = -4 \][/tex]
### Step 5: Substitute \(y = -4\) back into the expression for \(x\).
Now we use the expression \(x = \frac{5y + 4}{2}\):
[tex]\[ x = \frac{5(-4) + 4}{2} \][/tex]
[tex]\[ x = \frac{-20 + 4}{2} \][/tex]
[tex]\[ x = \frac{-16}{2} \][/tex]
[tex]\[ x = -8 \][/tex]
### Final Solution:
The solution to the system of equations is \(x = -8\) and \(y = -4\).
Therefore, the correct answer choice is:
[tex]\[ (-8, -4) \][/tex]
[tex]\[ \begin{array}{r} 2x = 5y + 4 \\ 3x - 2y = -16 \end{array} \][/tex]
Let's follow a step-by-step process to solve this system.
### Step 1: Express one variable in terms of the other from the first equation.
From the first equation:
[tex]\[ 2x = 5y + 4 \][/tex]
We can solve for \(x\):
[tex]\[ x = \frac{5y + 4}{2} \][/tex]
### Step 2: Substitute this expression into the second equation.
Substitute \(x = \frac{5y + 4}{2}\) into the second equation, \(3x - 2y = -16\):
[tex]\[ 3 \left( \frac{5y + 4}{2} \right) - 2y = -16 \][/tex]
### Step 3: Simplify the equation.
Multiply through by 2 to clear the fraction:
[tex]\[ 3(5y + 4) - 4y = -32 \][/tex]
[tex]\[ 15y + 12 - 4y = -32 \][/tex]
[tex]\[ 11y + 12 = -32 \][/tex]
### Step 4: Solve for \(y\).
Isolate \(y\) by subtracting 12 from both sides:
[tex]\[ 11y = -44 \][/tex]
Divide by 11:
[tex]\[ y = -4 \][/tex]
### Step 5: Substitute \(y = -4\) back into the expression for \(x\).
Now we use the expression \(x = \frac{5y + 4}{2}\):
[tex]\[ x = \frac{5(-4) + 4}{2} \][/tex]
[tex]\[ x = \frac{-20 + 4}{2} \][/tex]
[tex]\[ x = \frac{-16}{2} \][/tex]
[tex]\[ x = -8 \][/tex]
### Final Solution:
The solution to the system of equations is \(x = -8\) and \(y = -4\).
Therefore, the correct answer choice is:
[tex]\[ (-8, -4) \][/tex]
