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Let's solve the given problems step by step.

### Given Points:
- \( A(-2, 3) \)
- \( B(4, 5) \)
- \( C(-2, 4) \)

### (a) Equation of the line passing through points \( A \) and \( B \)

To find the equation of the line passing through two points, we use the formula for the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For points \( A(-2, 3) \) and \( B(4, 5) \):
\[ m = \frac{5 - 3}{4 - (-2)} = \frac{2}{6} = \frac{1}{3} \]

Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]

Using point \( A(-2, 3) \):
\[ y - 3 = \frac{1}{3}(x - (-2)) \]
\[ y - 3 = \frac{1}{3}(x + 2) \]
\[ y - 3 = \frac{1}{3}x + \frac{2}{3} \]
\[ y = \frac{1}{3}x + \frac{2}{3} + 3 \]
\[ y = \frac{1}{3}x + \frac{2}{3} + \frac{9}{3} \]
\[ y = \frac{1}{3}x + \frac{11}{3} \]

So, the equation of the line passing through points \( A \) and \( B \) is:
\[ y = \frac{1}{3}x + \frac{11}{3} \]

### (b) Equation of the line passing through points \( A \) and \( C \)

For points \( A(-2, 3) \) and \( C(-2, 4) \):

Since both points have the same x-coordinate, the line is vertical:
\[ x = -2 \]

### (c) Equation of the line passing through points \( B \) and \( C \)

For points \( B(4, 5) \) and \( C(-2, 4) \):
\[ m = \frac{4 - 5}{-2 - 4} = \frac{-1}{-6} = \frac{1}{6} \]

Using the point-slope form:
\[ y - y_1 = m(x - x_1) \]

Using point \( B(4, 5) \):
\[ y - 5 = \frac{1}{6}(x - 4) \]
\[ y - 5 = \frac{1}{6}x - \frac{4}{6} \]
\[ y - 5 = \frac{1}{6}x - \frac{2}{3} \]
\[ y = \frac{1}{6}x - \frac{2}{3} + 5 \]
\[ y = \frac{1}{6}x - \frac{2}{3} + \frac{15}{3} \]
\[ y = \frac{1}{6}x - \frac{2}{3} + 5 \]
\[ y = \frac{1}{6}x + \frac{13}{3} \]

So, the equation of the line passing through points \( B \) and \( C \) is:
\[ y = \frac{1}{6}x + \frac{13}{3} \]

### (d) Graph all three equations on one graph

To plot these equations, we need a coordinate system. Here are the equations we will graph:
1. \( y = \frac{1}{3}x + \frac{11}{3} \)
2. \( x = -2 \)
3. \( y = \frac{1}{6}x + \frac{13}{3} \)

I can help you visualize this using a graph plotting tool. 

Would you like me to generate the graph for you?

Ditetukan tiga titik A(-2, 3), B(4, 5) dan titik C(-2, 4). (a) Carilah persamaan garis yang melalui titik A dan B. (b) Carilah persamaan garis yang melalui titik A dan C. (c) Carilah persamaan garis yang melalui titik B dan C. (d) Buatlah grafiknya dalam satu gambar.

Learn how to find the equations of lines passing through points A(-2, 3), B(4, 5), and C(-2, 4). This problem covers linear equations, the slope formula, and the point-slope form, making it suitable for Grade 10 students. The problem includes a graph plotting all equations in one coordinate system.

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