Solve the system of equations:

[tex]\[
\begin{array}{l}
2.5y + 3x = 27 \\
5x - 2.5y = 5
\end{array}
\][/tex]

What equation is the result of adding the two equations?
[tex]\[ \square \][/tex]

What is the solution to the system?
[tex]\[ \square \][/tex]



Answer :

To solve the given system of equations:

[tex]\[ \begin{array}{l} 2.5 y + 3 x = 27 \\ 5 x - 2.5 y = 5 \end{array} \][/tex]

### Step 1: Adding the Two Equations

First, we add the two equations together:

[tex]\[ (2.5 y + 3 x) + (5 x - 2.5 y) = 27 + 5 \][/tex]

Let's combine like terms:

[tex]\[ 2.5 y - 2.5 y + 3 x + 5 x = 32 \][/tex]

[tex]\[ 0 + 8 x = 32 \][/tex]

[tex]\[ 8 x = 32 \][/tex]

So, the resulting equation from adding the two equations is:

[tex]\[ 8 x = 32 \][/tex]

### Step 2: Solving for [tex]\( x \)[/tex]

Now, solve for [tex]\( x \)[/tex]:

[tex]\[ x = \frac{32}{8} = 4 \][/tex]

### Step 3: Solving for [tex]\( y \)[/tex]

Next, substitute [tex]\( x = 4 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Let's use the first equation:

[tex]\[ 2.5 y + 3(4) = 27 \][/tex]

[tex]\[ 2.5 y + 12 = 27 \][/tex]

Subtract 12 from both sides:

[tex]\[ 2.5 y = 27 - 12 \][/tex]

[tex]\[ 2.5 y = 15 \][/tex]

Solve for [tex]\( y \)[/tex]:

[tex]\[ y = \frac{15}{2.5} = 6 \][/tex]

### Final Solution

Thus, the resulting equation from adding the two equations is:

[tex]\[ 8 x = 32 \][/tex]

And the solution to the system of equations is:

[tex]\[ x = 4 \quad \text{and} \quad y = 6 \][/tex]