Question

rearrange the equation so x is the independent variable. 6x + y = 4x + 11y y =

Explanation:

Step1: Subtract 4x and y from both sides

To isolate the terms with \( y \) on one side and terms with \( x \) on the other, we subtract \( 4x \) and \( y \) from both sides of the equation \( 6x + y = 4x + 11y \).
\[
6x + y - 4x - y = 4x + 11y - 4x - y
\]

Step2: Simplify both sides

Simplifying the left side: \( 6x - 4x + y - y = 2x \).
Simplifying the right side: \( 4x - 4x + 11y - y = 10y \). Wait, no, actually, after step 1, we should have:
Left side: \( 6x - 4x + y - y = 2x \)
Right side: \( 4x - 4x + 11y - y = 10y \)? No, that's incorrect. Wait, let's redo step 1 correctly. We want to get all \( x \) terms on one side and \( y \) terms on the other. So subtract \( 4x \) from both sides and subtract \( 11y \) from both sides? Wait, no, original equation: \( 6x + y = 4x + 11y \). Let's subtract \( 4x \) from both sides: \( 6x - 4x + y = 11y \), which is \( 2x + y = 11y \). Then subtract \( y \) from both sides: \( 2x = 11y - y \), which is \( 2x = 10y \)? No, that's not right. Wait, no, to solve for \( y \) in terms of \( x \), we can collect like terms. Let's move all \( x \) terms to the left and \( y \) terms to the right. So subtract \( 4x \) from both sides: \( 6x - 4x + y = 11y \), so \( 2x + y = 11y \). Then subtract \( y \) from both sides: \( 2x = 11y - y \), so \( 2x = 10y \)? No, that's incorrect. Wait, no, the correct way is: starting with \( 6x + y = 4x + 11y \), subtract \( 4x \) from both sides: \( 2x + y = 11y \), then subtract \( 11y \) from both sides: \( 2x + y - 11y = 0 \), so \( 2x - 10y = 0 \), then add \( 10y \) to both sides: \( 2x = 10y \), then divide both sides by 10: \( y = \frac{2x}{10} \), which simplifies to \( y = \frac{x}{5} \). Wait, let's do it step by step correctly.

Starting over:

Equation: \( 6x + y = 4x + 11y \)

Step 1: Subtract \( 4x \) from both sides:

\( 6x - 4x + y = 11y \)

Simplify left side: \( 2x + y = 11y \)

Step 2: Subtract \( 11y \) from both sides:

\( 2x + y - 11y = 0 \)

Simplify: \( 2x - 10y = 0 \)

Step 3: Add \( 10y \) to both sides:

\( 2x = 10y \)

Step 4: Divide both sides by 10:

\( y = \frac{2x}{10} \)

Simplify the fraction: \( \frac{2x}{10} = \frac{x}{5} \)

Wait, that's correct. Alternatively, let's do it by moving \( y \) terms to one side and \( x \) terms to the other.

Original equation: \( 6x + y = 4x + 11y \)

Subtract \( 4x \) from both sides: \( 2x + y = 11y \)

Subtract \( 11y \) from both sides: \( 2x = 10y \)

Then divide both sides by 10: \( y = \frac{2x}{10} = \frac{x}{5} \)

Yes, that's correct.

Answer:

\( y = \frac{1}{5}x \) (or \( y = \frac{x}{5} \))