Questions: 8-(3y+4x)=6(x-y) and 3y+2=8+10x

Transcript text: \[ 8-(3 y+4 x)=6(x-y) \text { and } 3 y+2=8+10 x \]

Solution

Solution Steps

To determine if the given lines are parallel, we need to compare their slopes. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope. We will convert both equations to this form and then compare the slopes.

Step 1: Convert Equations to Slope-Intercept Form

We start with the given equations: \[ 8 - (3y + 4x) = 6(x - y) \] \[ 3y + 2 = 8 + 10x \]

Step 2: Simplify the First Equation

Rewriting the first equation: \[ 8 - 3y - 4x = 6x - 6y \] Combine like terms: \[ -4x - 3y + 8 = 6x - 6y \] Rearrange to isolate \( y \): \[ -4x + 6x - 3y + 6y = -8 \] \[ 2x + 3y = 8 \] Solve for \( y \): \[ 3y = -2x + 8 \] \[ y = \frac{-2}{3}x + \frac{8}{3} \]

Step 3: Simplify the Second Equation

Rewriting the second equation: \[ 3y + 2 = 10x + 8 \] Rearrange to isolate \( y \): \[ 3y = 10x + 6 \] \[ y = \frac{10}{3}x + 2 \]

Step 4: Compare the Slopes

From the slope-intercept forms, we have: \[ y = \frac{-2}{3}x + \frac{8}{3} \] \[ y = \frac{10}{3}x + 2 \] The slopes are: \[ m_1 = \frac{-2}{3} \] \[ m_2 = \frac{10}{3} \]

Final Answer

Since the slopes \( m_1 \) and \( m_2 \) are not equal, the lines are not parallel.

\(\boxed{\text{No}}\)

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