Question

Consider the following system of equations: y−2x=−6 2y = x+3

Answer 1

To solve the system of equations given by:

1. \( y - 2x = -6 \)
2. \( 2y = x + 3 \)

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 \( y \):

1. \( y = 2x - 6 \)
2. \( y = \frac{1}{2}x + \frac{3}{2} \)

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.