Question

Write the equation in slope-intercept form that has a x-intercept of 30 and a y-intercept of 15

Answer 1

`\displaystyle y = -(1)/(2)\, x + 15`.

Explanation:

All non-horizontal line in the cartesian plane intersects the
`x`-axis at a unique point. The
`x\!`-coordinate of that point is the
`\! x`-intercept of this line.

The
`x`-intercept of the line in this question is
`30`. Thus, the
`x\!`-coordinate of the intersection of this line and the
`\! x`-axis would be
`30\!`.

Like all other points on the
`x`-axis, the
`y\!`-coordinate of that intersection would be
`0`. Therefore, the coordinates of that intersection would be
`(30,\, 0)`.

Similarly, the
`y`-intercept of a non-vertical line is the
`y\!`-coordinate of the point where that line intersects the
`y\!\!`-axis.

The slope-intercept form of a line is in the form
`y = m\, x + b`, where
`m` is the slope of the line and
`b` is the
`y`-intercept of this line. Both
`m\!` and
`b\!` are constants.

It is given that the
`y`-intercept of the line in this question is
`15`. Therefore,
`b = 15`. The slope-intercept equation of this line would be
`y = m\, x + 15` for some slope
`m` to be found.

All points
`(x,\, y)` on this line should satisfy the equation
`y = m\, x + 15` of this line. The
`x`-intercept of this line,
`(30,\, 0)`, is a point on this line. Thus, the equation
`y = m\, x + 15\!` should hold for
`x = 30` and
`y = 0`. Substitute these two values into the equation and solve for the slope
`m`:

`0 = 30\, m + 15`.

`\displaystyle m = -(1)/(2)`.

Therefore, the slope-intercept equation of this line would be:

`\displaystyle y = -(1)/(2)\, x + 15`.