Question

a line passes through the point (8, -4) and has a slope of \\(\frac{3}{4}\\). write an equation in slope-intercept form for this line.

Explanation:

Step1: Recall slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We know that the slope $m=\frac{3}{4}$. So we can start with the equation $y=\frac{3}{4}x + b$.

Step2: Substitute the point into the equation

We know that the line passes through the point $(8,-4)$. This means that when $x = 8$ and $y=-4$, the equation $y=\frac{3}{4}x + b$ holds. Substitute $x = 8$ and $y=-4$ into the equation:

$$ -4=\frac{3}{4}(8)+b $$

Step3: Simplify and solve for b

First, simplify $\frac{3}{4}(8)$. $\frac{3}{4}\times8 = 3\times2=6$. So the equation becomes:

$$ -4 = 6 + b $$

Subtract 6 from both sides of the equation to solve for $b$:

$$ b=-4 - 6=-10 $$

Step4: Write the final equation

Now that we know $m = \frac{3}{4}$ and $b=-10$, substitute these values into the slope - intercept form $y=mx + b$. We get:

$$ y=\frac{3}{4}x-10 $$

Answer:

$y=\frac{3}{4}x - 10$