IDNLearn.com is the perfect place to get detailed and accurate answers to your questions. Get comprehensive and trustworthy answers to all your questions from our knowledgeable community members.

A system of equations is shown:

[tex]\[
\begin{array}{l}
2x = 5y + 4 \\
3x - 2y = -16
\end{array}
\][/tex]

What is the solution to this system of equations?

A. \((-8, -4)\)
B. \((8, 4)\)
C. \((-4, -8)\)
D. [tex]\((4, 8)\)[/tex]


Sagot :

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]