Question

Find the equation of the line with slope −5 and that contains the point (−9,−5). Write the equation in the form y=mx+b and identify m and b. Find the equation of the line that contains the points (2,4) and (10,2). Write the equation in the form y=mx+b and identify m and b.

Answer 1

Answer & Step-by-step explanation:

Slope-intercept form:

$y=mx+b$

m is the slope and b is the y-intercept. Insert the given slope:

$y=-5x+b$

To find the y-intercept, take the given coordinate point and insert:

$(-9_{x},-5_{y})\\\\-5=-5(-9)+b$

Solve for b:

Simplify multiplication:

$-5=45+b$

Subtract 45 from both sides:

$-50=b\\\\b=-50$

The y-intercept is -50. Insert:

$y=-5x-50$

$m=-5\\b=-50$

Use the slope formula for when you have two points:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{rise}{run}$

The slope is the change in the y-axis over the change in the x-axis, or rise over run. Insert the points:

$(2_{x1},4_{y1})\\(10_{x2},2_{y2})$

$\frac{2-4}{10-2}=\frac{-2}{8}=-\frac{2}{8}=-\frac{1}{4}$

The slope is $-\frac{1}{4}$ . Insert:

$y=-\frac{1}{4}x+b$

Now follow the steps from the last problem to find the y-intercept. Choose a point and insert into the equation:

$(2_{x},4_{y})\\\\4=-\frac{1}{4}(2)+b$

Solve for b:

Simplify multiplication:

$-\frac{1}{4}*\frac{2}{1}=-\frac{2}{4}=-\frac{1}{2}$

Re-insert:

$4=-\frac{1}{2} +b$

Subtract b from both sides:

$4-b=-\frac{1}{2} +b-b\\\\4-b=-\frac{1}{2}$

Subtract 4 from both sides:

$4-4-b=-\frac{1}{2}-4\\\\-b=-\frac{1}{2}-4$

Simplify subtraction:

$-\frac{1}{2}-4=-\frac{1}{2} -\frac{4}{1} =-\frac{1}{2} -\frac{8}{2} \\\\-\frac{1}{2} -\frac{8}{2}=-\frac{9}{2}$

Re-insert:

$-b=-\frac{9}{2}$

Divide both sides by -1 to make the variable positive (can be seen as -1b):

$b=\frac{9}{2}$

The y-intercept is $\frac{9}{2}$ .

Write the equation:

$y=-\frac{1}{4}x+\frac{9}{2}$

:Done