$x$ -40 50 80 110

$y$ -40 -17.5 -10 -2.5

Tell whether each statement about line $j$ is True or False.

AThe slope of line $j$ is negative. [ ] True [ ] False
BThe $y$-intercept is negative. [ ] True [ ] False
CLine $j$ does not pass through $(-10,15)$. [ ] True [ ] False
DLine $j$ represents a set of all solutions to an equation in two variables. [ ] True [ ] False

Write an equation to represent line $j$.

Label two points on line $j$ and graph line $j$ in the coordinate plane.

To analyze line $j$, we use the provided data points: $(-40, -40)$, $(50, -17.5)$, $(80, -10)$, and $(110, -2.5)$. A line follows the equation $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. The slope is calculated as the ratio of the change in $y$ to the change in $x$:
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-17.5 - (-40)}{50 - (-40)} = \frac{22.5}{90} = 0.25 = \frac{1}{4}.\]
Using this slope, we find the y-intercept by substituting one point, for instance $(-40, -40)$, into $y = mx + c$:
\[-40 = 0.25(-40) + c \implies -40 = -10 + c \implies c = -30.\]
Thus, the equation of the line is $y = 0.25x - 30$. We can verify this with the other points, such as $(80, -10)$: $0.25(80) - 30 = 20 - 30 = -10$, which is correct.

Evaluating Statements

Now we evaluate the statements provided about line $j$:

AThe slope is $m = 0.25$. Since $0.25 > 0$, the slope is positive, not negative. Statement (a) is False.

BThe y-intercept is $c = -30$. Since $-30 < 0$, the y-intercept is indeed negative. Statement (b) is True.

CTo see if the line passes through $(-10, 15)$, we check the equation $y = 0.25x - 30$ at $x = -10$:

$y = 0.25(-10) - 30 = -2.5 - 30 = -32.5$.
Since $-32.5 eq 15$, the line does not pass through $(-10, 15)$. Statement (c) is True.

DA line in a coordinate plane is defined as the set of all points $(x, y)$ that satisfy its linear equation. Statement (d) is True.

Final Conclusion and Points

The equation for line $j$ is $y = \frac{1}{4}x - 30$. Two points on the line are $(-40, -40)$ and $(80, -10)$. These points can be plotted on a coordinate system to visualize the line, which starts in the third quadrant and rises towards the first quadrant, crossing the y-axis at $(0, -30)$.

Examples

Lines are fundamental in real-life applications for modeling linear relationships. For example, if a company charges a base fee of $\$30$ for a service and $\$0.25$ for every minute used, the total cost $y$ for $x$ minutes is given by the equation $y = 0.25x - 30$ (in a simplified model where the fee is discounted). Understanding these equations allows people to predict costs, track progress in data trends, and analyze the relationship between two changing quantities efficiently.