Question

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

Answer 1

Answer:

$\displaystyle y = -\frac{1}{2}\, x + 15$.

Step-by-step 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 $m$:

$0 = 30\, m + 15$.

$\displaystyle m = -\frac{1}{2}$.

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

$\displaystyle y = -\frac{1}{2}\, x + 15$.