Question

(5, 1) and (15, 16) are on the same line. What is the slope of the line? What is the y-intercept? Write the equation for the line.​

Answer 1

Explanation:

First, figure the rate of change [slope] out:

`\displaystyle (-y_1 + y_2)/(-x_1 + x_2) = m \\ \\ (-1 + 16)/(-5 + 15) = (15)/(10) = 1(1)/(2) \\ \boxed{1(1)/(2) = m}`

Now locate the initial value [y-intercept] by plugging an ordered pair into the slope-intercept formula. It does not matter which ordered pair you use:

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`\displaystyle y = mx + b`

16 = 1½[15] + b >>
`\displaystyle 16 = 22(1)/(2) + b; -6(1)/(2) = b`

OR

1 = 1½[5] + b >>
`\displaystyle 1 = 7(1)/(2) + b; -6(1)/(2) = b`

`\displaystyle [0, -6(1)/(2)]`

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You now have your initial value. All that is left is to write the equation of this line, which can be written two ways:

> Slope-Intercept Form

`\displaystyle y = 1(1)/(2)x - 6(1)/(2)`

> Standard Form
`\displaystyle [Ax + By = C]`

y = 1½x - 6½

- 1½x - 1½x

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–1½x + y = –6½ [We CANNOT leave the equation this way, so multiply by –2 to eradicate the fraction\denominatour (if you wrote the equation as mixed numbers\if you wrote the equation as improper fractions).]

–2[–1½x + y = –6½]

`\displaystyle 3x - 2y = 13`

With that, you have defined all the information.

`\displaystyle -3x + 2y = -13`

*About this equation, INSTEAD of multiplying by –2, you multiply by its oppocite, 2. Now, you can leave it like this, but UNIVERSALLY, the A-term is positive, so you must multiply the negative out as well.

I am joyous to assist you at any time.