To solve the system of equations given by:
- \( y - 2x = -6 \)
- \( 2y = x + 3 \)
we can start by rearranging each equation to express them in a more standard form.
Step 1: Rearranging the equations
From the first equation \( y - 2x = -6 \), we can isolate \( y \):
\[ y = 2x - 6 \]
From the second equation \( 2y = x + 3 \), we can isolate \( y \) as well:
\[ y = \frac{1}{2}x + \frac{3}{2} \]
Step 2: Setting the equations equal to each other
Now we have two expressions for \( y \):
- \( y = 2x - 6 \)
- \( y = \frac{1}{2}x + \frac{3}{2} \)
We can set these equal to solve for \( x \):
\[ 2x - 6 = \frac{1}{2}x + \frac{3}{2} \]
Step 3: Solving for \( x \)
To eliminate the fraction, we can multiply every term by 2:
\[ 2(2x - 6) = 2\left(\frac{1}{2}x + \frac{3}{2}\right) \]
This simplifies to:
\[ 4x - 12 = x + 3 \]
Now, we can isolate \( x \) by moving \( x \) to the left and \( -12 \) to the right:
\[ 4x - x = 3 + 12 \]
\[ 3x = 15 \]
Dividing by 3 gives:
\[ x = 5 \]
Step 4: Solving for \( y \)
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). Using the first equation:
\[ y = 2(5) - 6 \]
\[ y = 10 - 6 = 4 \]
Final Solution
The solution to the system of equations is:
\[ (x, y) = (5, 4) \]
We can confirm this solution by substituting \( x = 5 \) and \( y = 4 \) into the second equation:
\[ 2(4) = 5 + 3 \] \[ 8 = 8 \]
Both equations are satisfied, so the solution \( (5, 4) \) is correct.