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The question in the image asks to determine the value of \( c \) in the line equation \( 3x + cy = 5 \), based on the following conditions:

6. **Tentukan nilai \( c \) yang mana garis \( 3x + cy = 5 \)**  
a) Melalui titik \( (3,1) \)  
b) Sejajar dengan sumbu-y (Parallel to the y-axis)  
c) Sejajar dengan garis \( 2x + y = -1 \)  
d) Tegak lurus dengan garis \( y - 2 = 3(x + 3) \)  

### Explanation for each condition:

#### a) Through the point \( (3,1) \)
Substitute \( x = 3 \) and \( y = 1 \) into the equation \( 3x + cy = 5 \):
\[
3(3) + c(1) = 5 \implies 9 + c = 5 \implies c = 5 - 9 = -4
\]
Thus, for the line to pass through \( (3,1) \), \( c = -4 \).

#### b) Parallel to the y-axis
A line parallel to the y-axis has the form \( x = k \), where \( k \) is a constant. In this case, for the equation \( 3x + cy = 5 \) to represent a line parallel to the y-axis, the \( y \)-term must vanish. Hence, \( c = 0 \).

#### c) Parallel to the line \( 2x + y = -1 \)
Rewrite \( 2x + y = -1 \) in slope-intercept form:
\[
y = -2x - 1
\]
The slope of this line is \( -2 \). For two lines to be parallel, their slopes must be equal. The slope of \( 3x + cy = 5 \) is \( -\frac{3}{c} \). Setting the slopes equal:
\[
-\frac{3}{c} = -2 \implies c = \frac{3}{2}
\]
Thus, for the lines to be parallel, \( c = \frac{3}{2} \).

#### d) Perpendicular to the line \( y - 2 = 3(x + 3) \)
First, simplify the given line equation:
\[
y - 2 = 3(x + 3) \implies y = 3x + 11
\]
The slope of this line is \( 3 \). For two lines to be perpendicular, the product of their slopes must be \( -1 \). The slope of \( 3x + cy = 5 \) is \( -\frac{3}{c} \). Therefore:
\[
-\frac{3}{c} \times 3 = -1 \implies \frac{9}{c} = 1 \implies c = 9
\]
Thus, for the lines to be perpendicular, \( c = 9 \).

### Summary of answers:
a) \( c = -4 \)  
b) \( c = 0 \)  
c) \( c = \frac{3}{2} \)  
d) \( c = 9 \)

Would you like more details on any part of the solution?

Here are 5 related questions:
1. How do you find the equation of a line passing through two points?
2. What is the formula for the slope of a line?
3. How do you determine if two lines are perpendicular?
4. What happens to the equation of a line when the slope is zero?
5. How do you convert a linear equation to slope-intercept form?

**Tip:** Remember that parallel lines have equal slopes, while perpendicular lines have slopes that multiply to \(-1\).

Tentukan nilai c yang mana garis 3x + cy = 5 berdasarkan kondisi-kondisi berikut:
a) Melalui titik (3,1)
b) Sejajar dengan sumbu-y (Parallel to the y-axis)
c) Sejajar dengan garis 2x + y = -1
d) Tegak lurus dengan garis y - 2 = 3(x + 3)

Learn how to determine the value of c in the line equation 3x + cy = 5, given different conditions such as passing through a point, being parallel or perpendicular to other lines. This problem covers linear equations, slopes, and geometry concepts for high school students in Grades 10-12.

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