Answer:
[tex]\displaystyle y = -\frac{1}{2}\, x + 15[/tex].
Step-by-step explanation:
All non-horizontal line in the cartesian plane intersects the [tex]x[/tex]-axis at a unique point. The [tex]x\![/tex]-coordinate of that point is the [tex]\! x[/tex]-intercept of this line.
The [tex]x[/tex]-intercept of the line in this question is [tex]30[/tex]. Thus, the [tex]x\![/tex]-coordinate of the intersection of this line and the [tex]\! x[/tex]-axis would be [tex]30\![/tex].
Like all other points on the [tex]x[/tex]-axis, the [tex]y\![/tex]-coordinate of that intersection would be [tex]0[/tex]. Therefore, the coordinates of that intersection would be [tex](30,\, 0)[/tex].
Similarly, the [tex]y[/tex]-intercept of a non-vertical line is the [tex]y\![/tex]-coordinate of the point where that line intersects the [tex]y\!\![/tex]-axis.
The slope-intercept form of a line is in the form [tex]y = m\, x + b[/tex], where [tex]m[/tex] is the slope of the line and [tex]b[/tex] is the [tex]y[/tex]-intercept of this line. Both [tex]m\![/tex] and [tex]b\![/tex] are constants.
It is given that the [tex]y[/tex]-intercept of the line in this question is [tex]15[/tex]. Therefore, [tex]b = 15[/tex]. The slope-intercept equation of this line would be [tex]y = m\, x + 15[/tex] for some slope [tex]m[/tex] to be found.
All points [tex](x,\, y)[/tex] on this line should satisfy the equation [tex]y = m\, x + 15[/tex] of this line. The [tex]x[/tex]-intercept of this line, [tex](30,\, 0)[/tex], is a point on this line. Thus, the equation [tex]y = m\, x + 15\![/tex] should hold for [tex]x = 30[/tex] and [tex]y = 0[/tex]. Substitute these two values into the equation and solve for the slope [tex]m[/tex]:
[tex]0 = 30\, m + 15[/tex].
[tex]\displaystyle m = -\frac{1}{2}[/tex].
Therefore, the slope-intercept equation of this line would be:
[tex]\displaystyle y = -\frac{1}{2}\, x + 15[/tex].
