Question

The sides of a rectangle are given as: 4x+3, 4x-y, x+6y, and 3x+1. Find x and y.

Answer 1

To find the values of \( x \) and \( y \) that define the sides of a rectangle, we know that opposite sides must be equal in length. Given the sides of the rectangle are:

1. \( a = 4x + 3 \)
2. \( b = 4x - y \)
3. \( c = x + 6y \)
4. \( d = 3x + 1 \)

We can establish the following equations based on the fact that opposite sides of a rectangle are equal:

\[
a = c \quad \text{(1)}
\]
\[
b = d \quad \text{(2)}
\]

This sets up the system of equations:

1. \( 4x + 3 = x + 6y \)
2. \( 4x - y = 3x + 1 \)

Let's solve these equations step by step.

### Step 1: Solve Equation (1)

Starting with the first equation:

\[
4x + 3 = x + 6y
\]

Subtract \( x \) from both sides:

\[
4x - x + 3 = 6y
\]
\[
3x + 3 = 6y
\]

Now, divide everything by 3:

\[
x + 1 = 2y
\]

Rearranging gives us:

\[
y = \frac{x + 1}{2} \quad \text{(3)}
\]

### Step 2: Solve Equation (2)

Now, let's address the second equation:

\[
4x - y = 3x + 1
\]

Subtract \( 3x \) from both sides:

\[
4x - 3x - y = 1
\]
\[
x - y = 1
\]

Rearranging gives us:

\[
y = x - 1 \quad \text{(4)}
\]

### Step 3: Substitute and Solve for \( x \)

Now, we have two expressions for \( y \): equation (3) and equation (4). We can set these equal to each other:

\[
\frac{x + 1}{2} = x - 1
\]

To eliminate the fraction, multiply both sides by 2:

\[
x + 1 = 2(x - 1)
\]

Expanding the right side:

\[
x + 1 = 2x - 2
\]

Subtract \( x \) from both sides:

\[
1 = x - 2
\]

Adding 2 to both sides gives us:

\[
x = 3
\]

### Step 4: Find \( y \) using \( x \)

Now that we have \( x \), we can find \( y \) using equation (4):

\[
y = x - 1
\]

Substituting \( x = 3 \):

\[
y = 3 - 1 = 2
\]

### Conclusion

The values of \( x \) and \( y \) are:

\[
\boxed{(3, 2)}
\]