(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:

[tex]\displaystyle 3x - 2y = 13\:or\:y = 1\frac{1}{2}x - 6\frac{1}{2} \\ [0, -6\frac{1}{2}] \\ 1\frac{1}{2} = m[/tex]

Step-by-step explanation:

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

[tex]\displaystyle \frac{-y_1 + y_2}{-x_1 + x_2} = m \\ \\ \frac{-1 + 16}{-5 + 15} = \frac{15}{10} = 1\frac{1}{2} \\ \boxed{1\frac{1}{2} = m}[/tex]

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:

_______________________________________________

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

16 = 1½[15] + b >> [tex]\displaystyle 16 = 22\frac{1}{2} + b; -6\frac{1}{2} = b[/tex]

OR

1 = 1½[5] + b >> [tex]\displaystyle 1 = 7\frac{1}{2} + b; -6\frac{1}{2} = b[/tex]

[tex]\displaystyle [0, -6\frac{1}{2}][/tex]

_______________________________________________

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

[tex]\displaystyle y = 1\frac{1}{2}x - 6\frac{1}{2}[/tex]

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

y = 1½x - 6½

- 1½x - 1½x

___________

–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]

[tex]\displaystyle 3x - 2y = 13[/tex]

With that, you have defined all the information.

[tex]\displaystyle -3x + 2y = -13[/tex]

*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.

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