Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the slope of the line containing the points (3,-2) and (1,6)

Short answer

The slope of the line is -4.

Step-by-step solution

- Recall the Slope Formula

The formula to find the slope ( m ) of a line passing through two points ( x1, y1 ) and ( x2, y2 ) is given by: \[ m = \frac{y2 - y1}{x2 - x1} \]

- Identify the Points

From the problem, identify the coordinates of the two points. Here, the points are ( 3, -2 ) and ( 1, 6 ).

- Substitute the Values

Substitute the coordinates ( x1 = 3, y1 = -2 ) and ( x2 = 1, y2 = 6 ) into the slope formula: \[ m = \frac{6 - (-2)}{1 - 3} \]

- Simplify the Expression

Simplify the numerator and the denominator: \[ m = \frac{6 + 2}{1 - 3} = \frac{8}{-2} \]

- Calculate the Slope

Complete the division to find the slope: \[ m = \frac{8}{-2} = -4 \]

Key concepts

line slope

When you come across the term 'line slope' in mathematics, it refers to the steepness or the incline of the line. Understanding how to calculate the slope is crucial in various math problems and real-life situations.
The slope is defined as the ratio of the vertical change to the horizontal change between two points on a line. In simpler terms, it's how much the line goes up or down for every unit it goes across.
The formula to find the slope between two points \( (x1, y1) \) and \( (x2, y2) \) is: \[ m = \frac{y2 - y1}{x2 - x1} \]
Here, \( m \) represents the slope. Let's take a practical example. Let's say we have two points: (3, -2) and (1, 6). We can plug these points into the slope formula:
\[ m = \frac{6 - (-2)}{1 - 3} = \frac{8}{-2} = -4 \]
This calculation shows us that the slope of the line passing through these points is -4. That means for every unit we move to the right, the line goes down by 4 units.

coordinate geometry

Coordinate geometry, also called analytic geometry, describes the use of a coordinate system to study geometry. This method involves using coordinates or ordered pairs \( (x, y) \) to denote points on a plane.
Through this, you can analyze various properties of geometric figures and perform calculations that elucidate the relationships between these figures.
In our current context, we use coordinate geometry to find the slope of a line connecting two points. The coordinates of the points provide the necessary numerical values to perform these calculations. For example, if you have the points (3, -2) and (1, 6), you can directly substitute these values into the slope formula. This approach combines algebra and geometry to solve problems efficiently.

algebra steps

Understanding and applying algebra steps is essential for solving mathematical problems, including finding the slope of a line. The algebraic process involves breaking down the given problem into smaller, manageable steps.
Here are the steps to calculate the slope of the line through the points (3, -2) and (1, 6):

• First, recall the slope formula: \( m = \frac{y2 - y1}{x2 - x1} \). This will be the base of our calculations.
• Next, identify the coordinates: \( (3, -2) \) and \( (1, 6) \). Label them \( x1, y1 \) and \( x2, y2 \).
• Substitute these values into the formula: \( m = \frac{6 - (-2)}{1 - 3} \).
• Simplify the expression: \[ m = \frac{6 + 2}{1 - 3} = \frac{8}{-2} \]

Finally, perform the division to determine the slope: \[ m = \frac{8}{-2} = -4 \]
By following these algebra steps methodically, you ensure an accurate calculation of the slope and a deeper understanding of the process.