Question

Consider the system of equations below:
-2x + y = 2
2x + 5y = 19

What is the solution to the system?

Answer Options:

(1/2, 4)
(1/2, 5)
(8/3, 2)
(1/3, 3)

Answer 1

To solve the system of equations given by:

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

we can use the substitution or elimination method. Here, I will use substitution.

First, we can solve Equation 1 for \(y\):

\[
y = 2 + 2x
\]

Now, we can substitute this expression for \(y\) into Equation 2:

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

Expanding the equation gives:

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

Combining like terms results in:

\[
12x + 10 = 19
\]

Next, subtract 10 from both sides:

\[
12x = 9
\]

Then divide by 12:

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

Now that we have \(x\), we can find \(y\) by substituting \(x\) back into the expression for \(y\):

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

Calculating this gives:

\[
y = 2 + \frac{6}{4} = 2 + \frac{3}{2} = 2 + 1.5 = 3.5
\]

So the solution to the system is:

\[
\left(\frac{3}{4}, \frac{7}{2}\right) \quad \text{(which is not listed in the options)}
\]

Now let's revisit and plug back the derived values into the equations to check against provided options.

After checking all provided options:

- None of the options provided — \((1/2, 4)\), \((1/2, 5)\), \((8/3, 2)\), \((1/3, 3)\) — are solutions for the equations obtained.

Thus, we can conclude that none of the answer options is correct for the current system of equations.

Let's therefore check the validity of the two equations separately in relation to the answer options to try to get correct values.

Let's work the second equation with our potential solutions:

If we try:

**Option (1/2, 4):**
\[2(1/2) + 5(4) = 1 + 20 = 21 \quad \text{(not a solution)}\]

**Option (1/2, 5):**
\[2(1/2) + 5(5) = 1 + 25 = 26 \quad \text{(not a solution)}\]

**Option (8/3, 2):**
\[2(8/3) + 5(2) = 16/3 + 10 = 16/3 + 30/3 = 46/3 \quad \text{(not a solution)}\]

**Option (1/3, 3):**
\[2(1/3) + 5(3) = 2/3 + 15 = 45/3 + 2/3 = 47/3 \quad \text{(not a solution)}\]

Thus, no solution provided; further analysis yields the correct pairing or adjustments are needed to align values. You may provide new settings or ask further if resolution is required reformulating inequalities or constructs for more exact approaches fitting good results.