Question

Consider the system of equations below.
{-2x+y=2
{2x+5y=19
What is the solution to this equation?

Answer 1

To solve the system of equations:

1. \(-2x + y = 2\) (Equation 1)
2. \(2x + 5y = 19\) (Equation 2)

We can use the substitution or elimination method. In this case, we will first solve for \(y\) in terms of \(x\) using Equation 1.

### Step 1: Solve for \(y\) in Equation 1

From Equation 1:

\[
y = 2 + 2x
\]

### Step 2: Substitute \(y\) in Equation 2

Now substitute \(y\) in Equation 2:

\[
2x + 5(2 + 2x) = 19
\]

Distributing the \(5\):

\[
2x + 10 + 10x = 19
\]

Combine like terms:

\[
12x + 10 = 19
\]

### Step 3: Solve for \(x\)

Now isolate \(12x\):

\[
12x = 19 - 10
\]
\[
12x = 9
\]

Divide both sides by \(12\):

\[
x = \frac{9}{12} = \frac{3}{4}
\]

### Step 4: Substitute \(x\) back to find \(y\)

Now substitute \(x = \frac{3}{4}\) back into the expression for \(y\):

\[
y = 2 + 2\left(\frac{3}{4}\right)
\]
\[
y = 2 + \frac{6}{4}
\]
\[
y = 2 + \frac{3}{2} = \frac{4}{2} + \frac{3}{2} = \frac{7}{2}
\]

### Conclusion

The solution to the system of equations is:

\[
\left(x, y\right) = \left(\frac{3}{4}, \frac{7}{2}\right)
\]