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

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.

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 $\frac{4}{3}$.

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 $(-3, 1)$.

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

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

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

Final Answer