Question

a line passes through the points (5, -7.5) and has a y-intercept 10. which line has a rate of change that is less than this lines rate of change?
○ $y = 0.5x + 10$
○ $y = -1.5x + 10$
○ $y = -3.5x + 10$
○ $y = -10x - 3.5$

Explanation:

Step1: Recall slope-intercept form

The slope-intercept form of a line is $y = mx + b$, where $m$ is the rate of change (slope), and $b$ is the y-intercept. We know the given line has $b=10$, so its equation is $y = mx + 10$.

Step2: Calculate slope of given line

Substitute the point $(5, -7.5)$ into $y = mx + 10$ to solve for $m$.
$$-7.5 = m(5) + 10$$
Subtract 10 from both sides:
$$-7.5 - 10 = 5m$$
$$-17.5 = 5m$$
Divide both sides by 5:
$$m = \frac{-17.5}{5} = -3.5$$

Step3: Compare slopes

We need a line with a rate of change (slope) less than $-3.5$. On the number line, numbers less than $-3.5$ are more negative.

• $0.5 > -3.5$
• $-1.5 > -3.5$
• $-3.5 = -3.5$
• $-10 < -3.5$

Answer:

D. $y = -10x - 3.5$