Question

consider the system of equations below

{-2x+y=2
{2x+5y=19
what is the solution to the system


(118,414)
open paren 1 and 1 eighth comma 4 and 1 fourth close paren

(123,513)
open paren 1 and 2 thirds comma 5 and 1 third close paren

(314,212)
open paren 3 and 1 fourth comma 2 and 1 half close paren

(34,312)

Answer 1

To solve the system of equations

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

we can use substitution or elimination. I'll solve for \(y\) in terms of \(x\) using the first equation, and then substitute that into the second equation.

From the first equation:

\[
y = 2 + 2x
\]

Now substitute \(y\) into the second equation:

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

Expand this:

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

Combine like terms:

\[
12x + 10 = 19
\]

Now, subtract 10 from both sides:

\[
12x = 9
\]

Now divide by 12:

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

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

\[
y = 2 + 2\left(\frac{3}{4}\right) = 2 + \frac{3}{2} = 2 + 1.5 = 3.5 = \frac{7}{2}
\]

Thus the solution to the system of equations is:

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

Converting to mixed numbers gives:

\[
\left( 3 \frac{1}{4}, 2 \frac{1}{2} \right)
\]

Now reviewing the options:

- \( (118, 414) \)
- \( (1 \frac{1}{8}, 4 \frac{1}{4}) \)
- \( (123, 513) \)
- \( (1 \frac{2}{3}, 5 \frac{1}{3}) \)
- \( (314, 212) \)
- \( (3 \frac{1}{4}, 2 \frac{1}{2}) \)
- \( (34, 312) \)

The correct option based on our solution is:

\[
(3 \frac{1}{4}, 2 \frac{1}{2})
\]