Slope of a line

Question 1: Calculating the Slope of a Line

The Question: Find the slope of a line passing through the origin and the point $(-3, -2)$.

• 4/6
• -2/3
• -1/3
• 3/2

Core Concept: The Slope Formula

The slope of a line (often denoted by the letter $m$) is a measure of its steepness. It’s calculated as the ratio of the vertical change (“rise”) to the horizontal change (“run”) between any two points on the line.

For two points, $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is given by the formula:

Detailed Solution:

1. Identify the two points:

   • The first point is the origin: $(x_1, y_1) = (0, 0)$.
   • The second point is given: $(x_2, y_2) = (-3, -2)$.

2. Apply the slope formula:

   $$m = \frac{-2 - 0}{-3 - 0}$$

3. Calculate the result:

   $$m = \frac{-2}{-3}$$

   A negative divided by a negative is a positive.

   $$m = \frac{2}{3}$$

4. Check the options: The first option is 4/6. If we simplify this fraction by dividing the numerator and denominator by 2, we get $\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$. This matches our result.

Final Answer: The slope is 2/3, which corresponds to the option 4/6.

Question 2: Finding an Unknown Coordinate Using Slope

The Question: If the slope of a line passing through $P(1, 0)$ and $Q(-2, k)$ is 1, then the value of $k$ is __________.

• -2
• -3
• 0
• 2

Core Concept: The Slope Formula (in reverse)

We use the same slope formula, but this time we know the slope ($m$) and need to solve for one of the missing coordinates. We’ll set up an algebraic equation.

Detailed Solution:

1. Identify the knowns:

   • Point 1: $(x_1, y_1) = (1, 0)$.
   • Point 2: $(x_2, y_2) = (-2, k)$.
   • The slope: $m = 1$.

2. Set up the equation using the slope formula:

   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
   $$1 = \frac{k - 0}{-2 - 1}$$

3. Solve the equation for k:

   $$1 = \frac{k}{-3}$$

   Multiply both sides by -3:

   $$1 \times (-3) = k$$
   $$-3 = k$$

Final Answer: The value of $k$ is -3.

Question 3: Properties of Slopes

The Question: Which of the following is/are correct with respect to the slope of a line? (Multiple Select Question)

• The line parallel to the X-axis has slope of value 1
• The slope of a line which is parallel to the Y-axis is not defined
• The line parallel to the Y-axis has a slope of 90
• The line parallel to the X-axis has a slope 0
• The slope of a line is always positive

Core Concepts: Slopes of Horizontal and Vertical Lines

• Horizontal Line (parallel to the x-axis): A horizontal line has no vertical change, or “rise”. Its rise is always 0. Therefore, its slope is $m = \frac{0}{\text{run}} = 0$.
• Vertical Line (parallel to the y-axis): A vertical line has no horizontal change, or “run”. Its run is always 0. Division by zero is mathematically impossible. Therefore, the slope of a vertical line is undefined.

Detailed Solution:

Let’s evaluate each statement:

1. “The line parallel to the X-axis has slope of value 1”: This is FALSE. A line parallel to the x-axis is horizontal and has a slope of 0.
2. “The slope of a line which is parallel to the Y-axis is not defined”: This is TRUE. A line parallel to the y-axis is vertical, and its slope is undefined.
3. “The line parallel to the Y-axis has a slope of 90”: This is FALSE. This confuses the angle of inclination (90 degrees) with the slope. The slope is undefined.
4. “The line parallel to the X-axis has a slope 0”: This is TRUE.
5. “The slope of a line is always positive”: This is FALSE. A line can have a positive slope (rising left to right), a negative slope (falling left to right), or a zero slope (horizontal).

Final Answer: The two correct statements are “The slope of a line which is parallel to the Y-axis is not defined” and “The line parallel to the X-axis has a slope 0”.

Question 4: Lines with Zero Slope

The Question: Through which of the following points do lines of slope 0 pass? (Multiple Select Question)

• (0, 0)
• (-100, 0.5)
• (-0.05, 0)
• (0, 4)

Core Concept: Horizontal Lines

A line with a slope of 0 is a horizontal line. The equation for any horizontal line is of the form $y = c$, where $c$ is a constant. This means that every single point on that line must have the exact same y-coordinate.

Detailed Solution:

The question can be interpreted as asking, “Which of these points could lie on the same horizontal line?” To answer this, we look for points that share the same y-coordinate.

1. Point (0, 0): The y-coordinate is 0.
2. Point (-100, 0.5): The y-coordinate is 0.5.
3. Point (-0.05, 0): The y-coordinate is 0.
4. Point (0, 4): The y-coordinate is 4.

By comparing the y-coordinates, we can see that the points (0, 0) and (-0.05, 0) both have a y-coordinate of 0. Therefore, they both lie on the same horizontal line, which is the x-axis ($y=0$), a line with a slope of 0. The other two points, (-100, 0.5) and (0, 4), lie on different horizontal lines ($y=0.5$ and $y=4$, respectively).

Final Answer: The points (0, 0) and (-0.05, 0) lie on the same line of slope 0.