Exercise 14 Page 175

Using the given points, $(4, 1)$ and $(8, 4),$ we will write an equation in slope-intercept form. However, we cannot determine the $y -$ intercept of the equation from the given points. Therefore, we will follow three steps.

First, we will find the slope of the equation by using the Slope Formula.
Next, we will write the equation in point-slope form.
Finally, we will rearrange the equation to write it in slope-intercept form.

Finding the Slope

We know that the line passes through the points

(4, 1)

and

(8, 4) .

Let's substitute these points into the Slope Formula and find the slope.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(4, 1)$ & $(8, 4)$

m = \frac{4 - 1}{8 - 4}

SubTerms

Subtract terms

m = \frac{3}{4}

The slope of the line is

\frac{3}{4} .

Point-Slope Form

We know the slope of the line and two points that are on the line. We can choose one of these points and write the equation of the line. Let's choose the point

(4, 1) .

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

Slope-Intercept Form

Finally, we can write the equation of the line in slope-intercept form by isolating the

y -

variable in the equation.

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

Distr

Distribute $\frac{3}{4}$

y - 1 = \frac{3}{4} x - 3

AddEqn

$LHS + 1 = RHS + 1$

y = \frac{3}{4} x - 2

Therefore, the

y -

variable is isolated and we have our equation in slope-intercept form.

y = \frac{3}{4} x - 2