Questions: Find the equation of the line through (-3,1) and parallel to 3y-4x=3 A y=(4 / 3) x+3 B y=(3 / 4) x-5 C y=(4 / 3) x+5 D none of the above

Transcript text: Find the equation of the line through $(-3,1)$ and parallel to $3 y-4 x=3$ A $y=(4 / 3) x+3$ B $y=(3 / 4) x-5$ C $y=(4 / 3) x+5$ D none of the above Show Answer

Solution

Solution Steps

To find the equation of a line parallel to a given line, we need to determine the slope of the given line. Lines that are parallel have the same slope. The given line is in the form $3y - 4x = 3$. We can rearrange this into the slope-intercept form $y = mx + b$ to find the slope. Once we have the slope, we use the point-slope form of a line equation with the given point $(-3, 1)$ to find the equation of the new line.

Step 1: Determine the Slope of the Given Line

The given line is $3y - 4x = 3$. To find the slope, we rearrange this equation into the slope-intercept form $y = mx + b$.

\[ 3y = 4x + 3 \implies y = \frac{4}{3}x + 1 \]

The slope $m$ of the given line is $\frac{4}{3}$.

Step 2: Use the Point-Slope Form to Find the New Line

Since the new line is parallel to the given line, it will have the same slope, $\frac{4}{3}$. We use the point-slope form of the equation of a line, $y - y_1 = m(x - x_1)$, with the point $(-3, 1)$.

\[ y - 1 = \frac{4}{3}(x + 3) \]

Step 3: Convert to Slope-Intercept Form

Simplify the equation to convert it into the slope-intercept form $y = mx + b$.

\[ y - 1 = \frac{4}{3}x + 4 \implies y = \frac{4}{3}x + 5 \]