Question

Consider the system of equations below.
{-2x+y=2
{2x+5y=19
What is the solution to this equation?
responses
A) (1 1/8,4 1/4)
open paren 1 and 1 eighth comma 4 and 1 fourth close paren

B) (1 2/3,5 1/3)
open paren 1 and 2 thirds comma 5 and 1 third close paren

C) (3 1/4,2 1/2)
open paren 3 and 1 fourth comma 2 and 1 half close paren

D) (3/4,3 1/2)

Answer 1

To solve the system of equations:

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

Let's first 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 this gives:

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

Combining like terms:

\[
12x + 10 = 19
\]

Subtracting 10 from both sides gives:

\[
12x = 9
\]

Dividing by 12:

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

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

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

Calculating that:

\[
y = 2 + \frac{6}{4} = 2 + \frac{3}{2} = \frac{4}{2} + \frac{3}{2} = \frac{7}{2}
\]

So, \(y = \frac{7}{2}\) or \(3.5\).

The solution to the system of equations is \(\left(\frac{3}{4}, \frac{7}{2}\right)\) or \(\left(\frac{3}{4}, 3.5\right)\).

Now let's compare this with the given options:

A) \((1 \frac{1}{8}, 4 \frac{1}{4})\) - \((1.125, 4.25)\)

B) \((1 \frac{2}{3}, 5 \frac{1}{3})\) - \((1.667, 5.333)\)

C) \((3 \frac{1}{4}, 2 \frac{1}{2})\) - \((3.25, 2.5)\)

D) \((\frac{3}{4}, 3 \frac{1}{2})\) - \((0.75, 3.5)\)

The only option that matches our solution \(\left(\frac{3}{4}, \frac{7}{2}\right)\) is:

**D) \((\frac{3}{4}, 3 \frac{1}{2})\)**.