Question

Solve the system of linear equations 2x + 5y = 7 and x - 2y = -3.

Answer 1

To solve the system of linear equations given by:

1. \( 2x + 5y = 7 \) (Equation 1)
2. \( x - 2y = -3 \) (Equation 2)

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

First, let's solve Equation 2 for \( x \):

\[
x = 2y - 3
\]

Now, we can substitute this expression for \( x \) into Equation 1:

\[
2(2y - 3) + 5y = 7
\]

Now, expand and simplify:

\[
4y - 6 + 5y = 7
\]

Combine like terms:

\[
(4y + 5y) - 6 = 7
\]

\[
9y - 6 = 7
\]

Now, add 6 to both sides:

\[
9y = 13
\]

Now, divide by 9:

\[
y = \frac{13}{9}
\]

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

\[
x = 2\left(\frac{13}{9}\right) - 3
\]

Calculating \( x \):

\[
x = \frac{26}{9} - 3
\]

To write 3 in terms of ninths, we have:

\[
3 = \frac{27}{9}
\]

So,

\[
x = \frac{26}{9} - \frac{27}{9} = \frac{26 - 27}{9} = \frac{-1}{9}
\]

Thus, the solution to the system of equations is:

\[
x = -\frac{1}{9}, \quad y = \frac{13}{9}
\]

In ordered pair form, the solution is:

\[
\left( -\frac{1}{9}, \frac{13}{9} \right)
\]