Question

Solve the system of linear equations: 2x + 5y = 11 and x - 2y = -3, and verify the solution by substituting the coordinates into both equations.

Answer 1

To solve the system of linear equations:

1. \( 2x + 5y = 11 \) (Equation 1)
2. \( x - 2y = -3 \) (Equation 2)

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

First, solve Equation 2 for \( x \):

\[
x = 2y - 3
\]

Now substitute \( x \) in Equation 1:

\[
2(2y - 3) + 5y = 11
\]

Now, distribute the 2:

\[
4y - 6 + 5y = 11
\]

Combine like terms:

\[
9y - 6 = 11
\]

Now, add 6 to both sides:

\[
9y = 17
\]

Divide by 9:

\[
y = \frac{17}{9}
\]

Now substitute \( y \) back into the expression for \( x \):

\[
x = 2\left(\frac{17}{9}\right) - 3
\]

Calculating \( x \):

\[
x = \frac{34}{9} - 3 = \frac{34}{9} - \frac{27}{9} = \frac{7}{9}
\]

So, the solution to the system of equations is:

\[
x = \frac{7}{9}, \quad y = \frac{17}{9}
\]

### Verification

Now we will verify our solution by substituting \( x \) and \( y \) back into both original equations.

1. Substituting into Equation 1:

\[
2\left(\frac{7}{9}\right) + 5\left(\frac{17}{9}\right) = 11
\]

Calculating:

\[
\frac{14}{9} + \frac{85}{9} = \frac{99}{9} = 11
\]

Equation 1 is satisfied.

2. Substituting into Equation 2:

\[
\left(\frac{7}{9}\right) - 2\left(\frac{17}{9}\right) = -3
\]

Calculating:

\[
\frac{7}{9} - \frac{34}{9} = -\frac{27}{9} = -3
\]

Equation 2 is satisfied.

Both equations are satisfied, confirming that the solution is correct.

Thus, the solution to the system of equations is:

\[
\boxed{\left(\frac{7}{9}, \frac{17}{9}\right)}
\]