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Slope-intercept form

Linear equations are often organized in slope-intercept form:

[tex]y=mx+b[/tex]

  • (x,y) = a point that falls on the line
  • m = the slope of the line
  • b = the y-intercept of the line

Slope (m)

The slope of a line is equal to its [tex]\dfrac{rise}{run}[/tex].

  • "Rise" refers to the number of units the line travels up.
  • "Run" refers to the number of units the line travels to the right.

Typically, we would solve for the slope by using the following formula:

  • [tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex] where two points that fall on the line are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]

Y-intercept (b)

The y-intercept of a line refers to the y-value that occurs when x=0.

On a graph, it is the y-value where the line crosses the y-axis.

Writing the Equation

1) Determine the slope of the line (m)

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

Plug in the two given points, (-17,-4) and (-7,-13):

[tex]m=\dfrac{-13-(-4)}{(-7)-(-17)}\\\\m=\dfrac{-13+4}{-7+17}\\\\m=\dfrac{-9}{10}[/tex]

Therefore, the slope of the line is [tex]-\dfrac{9}{10}[/tex]. Plug this into [tex]y=mx+b[/tex]:

[tex]y=-\dfrac{9}{10}x+b[/tex]

2) Determining the y-intercept (b)

[tex]y=-\dfrac{9}{10}x+b[/tex]

Plug in one of the given points and solve for b:

[tex]-4=-\dfrac{9}{10}(-17)+b\\\\-4=\dfrac{153}{10}+b\\\\b=-\dfrac{193}{10}[/tex]

Therefore, the y-intercept of the line is [tex]-\dfrac{193}{10}[/tex]. Plug this back into our equation:

[tex]y=-\dfrac{9}{10}x-\dfrac{193}{10}[/tex]

Answer

[tex]y=-\dfrac{9}{10}x-\dfrac{193}{10}[/tex]