Let's solve the even-numbered exercises from the provided image.

Numeric Section

38. $4x - 3y = 12$

To find the x-intercept, set $y = 0$: $4x = 12 \implies x = 3$ The x-intercept is $(3, 0)$.

To find the y-intercept, set $x = 0$: $-3y = 12 \implies y = -4$ The y-intercept is $(0, -4)$.

The points to plot are $(3, 0)$ and $(0, -4)$.

40. $y - 5 = 5x$

First, rewrite it in slope-intercept form $y = mx + b$: $y = 5x + 5$

To find the x-intercept, set $y = 0$: $0 = 5x + 5 \implies 5x = -5 \implies x = -1$ The x-intercept is $(-1, 0)$.

To find the y-intercept, set $x = 0$: $y = 5$ The y-intercept is $(0, 5)$.

The points to plot are $(-1, 0)$ and $(0, 5)$.

42. $y = -\frac{7}{3} x$

To find the x-intercept, set $y = 0$: $0 = -\frac{7}{3} x \implies x = 0$ The x-intercept is $(0, 0)$, which is also the y-intercept.

Graph Questions Section

44. Find the coordinates of the midpoint of the line segment connecting the two points.

Given the endpoints are $(2, 3)$ and $(5, 1)$, the midpoint formula is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ $M = \left( \frac{2 + 5}{2}, \frac{3 + 1}{2} \right)$ $M = \left( \frac{7}{2}, \frac{4}{2} \right)$ $M = \left( 3.5, 2 \right)$ So, the midpoint is $(3.5, 2)$.

46. Find the distance that $(5, 2)$ is from the origin. Round to three decimal places.

The distance formula is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ For the point $(5, 2)$ and the origin $(0, 0)$: $d = \sqrt{(5 - 0)^2 + (2 - 0)^2}$ $d = \sqrt{5^2 + 2^2}$ $d = \sqrt{25 + 4}$ $d = \sqrt{29} \approx 5.385$ So, the distance is approximately $5.385$.

Would you like more details or have any questions? Here are some related questions:

1. How do you find the slope of a line given two points?
2. What is the significance of the slope-intercept form of a line equation?
3. How do you graph a line using its x- and y-intercepts?
4. How do you find the distance between two points on a coordinate plane?
5. What are some real-world applications of finding midpoints?
6. How can you verify the solution to a linear equation?
7. What is the relationship between the distance formula and the Pythagorean theorem?
8. How do you convert between different forms of a linear equation?

Tip: When graphing a line, always double-check the intercepts by substituting them back into the original equation to ensure accuracy.