https://file.aiquickdraw.com/mathbot/file/1722952286266qwx4ls0d.jpg

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.

Explore how to solve coordinate geometry problems involving midpoints and distances. This page covers finding midpoints and distances between points on a coordinate plane using the midpoint formula and distance formula. Suitable for students in Grades 7-9.

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