Sure, let's solve the problem step by step for each line:
### 1. For the line [tex]\(x = 3\)[/tex]:
- This is a vertical line because it only specifies the x-coordinate and does not change with y.
- The slope of a vertical line is undefined.
Answer: Undefined slope
### 2. For the line [tex]\(6x + 3y = 18\)[/tex]:
- First, convert the given equation to the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
6x + 3y = 18
\][/tex]
Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
3y = -6x + 18
\][/tex]
Divide by 3:
[tex]\[
y = -2x + 6
\][/tex]
- The slope (-2) is the coefficient of [tex]\(x\)[/tex].
Answer: -2
### 3. For the line passing through [tex]\((7,5)\)[/tex] and [tex]\((-8,5)\)[/tex]:
- Use the slope formula:
[tex]\[
\text{slope} = \frac{y2 - y1}{x2 - x1}
\][/tex]
Here, [tex]\((x1, y1) = (7, 5)\)[/tex] and [tex]\((x2, y2) = (-8, 5)\)[/tex]:
[tex]\[
\text{slope} = \frac{5 - 5}{-8 - 7} = \frac{0}{-15} = 0
\][/tex]
- When the y-coordinates are the same, the slope is 0.
Answer: 0
### 4. For the line passing through [tex]\((4,2)\)[/tex] and [tex]\((1,-7)\)[/tex]:
- Use the slope formula:
[tex]\[
\text{slope} = \frac{y2 - y1}{x2 - x1}
\][/tex]
Here, [tex]\((x1, y1) = (4,2)\)[/tex] and [tex]\((x2, y2) = (1, -7)\)[/tex]:
[tex]\[
\text{slope} = \frac{-7 - 2}{1 - 4} = \frac{-9}{-3} = 3
\][/tex]
- The slope is 3.
Answer: 3
Now we can match the lines with their corresponding slopes:
[tex]\[
\begin{array}{l}
x=3 & \text{Undefined slope} \\
6x + 3y=18 & -2 \\
\text{The line through } (7,5) \text{ and } (-8,5) & 0 \\
\text{The line through } (4,2) \text{ and } (1,-7) & 3 \\
\end{array}
\][/tex]
