Question

Write the equation of each line in slope-intercept form.
(If possible please show work)

Answer 1

`\displaystyle y = -(2)/(3)\, x - 9`.

Explanation:

The slope-intercept form of a line on a cartesian plane should be in the form:

`y = m\, x + b`,

where:

• 
  `m` is the slope of the line, while
• 
  `b` is the
  `y`-coordinate of the point where the line intersects the
  `y`-axis.

The question states that the slope of this line is
`\displaystyle -(2)/(3)`. In other words,
`\displaystyle m = -(2)/(3)`. The next step is to find the value of
`b`. That could be done using the information that the point
`(-6,\, -5)` is on this line.

Note that the slope-intercept form of a line
`y = m\, x + b` is essentially an equation about
`x` and
`y`. For a point
`(x_0,\, y_0)` to be on that line,
`x = x_0` and
`y = y_0` should satisfy its equation. In other words, it must be true that
`y_0 = m\, x_0 + b`.

For the point
`(-6,\, -5)`,
`x_0 = -6` and
`y_0 = -5`. The equation would be:

`\underbrace{-5}_(y_0) = m * \underbrace{(-6)}_(x_0)+ b`.

Besides, the slope of this line is already known to be
`\displaystyle m = -(2)/(3)`. Therefore, this equation would become:

`\displaystyle \underbrace{-5}_(y_0) = \underbrace{\left(-(2)/(3)\right)}_(m) * \underbrace{(-6)}_(x_0)+ b`.

Solve this equation for
`b`:

`b = -9`.

Hence, the slope-intercept form (
`y = m\, x + b`) of this line would be:

`\displaystyle y = -(2)/(3)\, x - 9`.