Question

consider the system of equations below
{-2x + y = 2
{2x + 5y = 19
what is the solution of the system

Answer 1

To solve the system of equations:

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

We can use the substitution or elimination method. Let's use the substitution method.

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

\[
y = 2 + 2x
\]

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

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

Distribute \(5\):

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

Combine like terms:

\[
12x + 10 = 19
\]

Subtract \(10\) from both sides:

\[
12x = 9
\]

Now divide by \(12\):

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

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

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

Calculate \(y\):

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

The solution to the system of equations is:

\[
x = \frac{3}{4}, \quad y = \frac{7}{2}
\]

So the solution is \(\left( \frac{3}{4}, \frac{7}{2} \right)\).