Problem 43 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. \((7,2),(5,6)\)

Step-by-Step Solution

Answer

The slope of the line is \(-2\).

Step 1: Understanding the Problem

To find the slope of the line passing through two points, we use the formula for the slope \( m \) of a line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, we identify \( (x_1, y_1) = (7, 2) \) and \( (x_2, y_2) = (5, 6) \).

Step 2: Substitute the Values

Substitute the values from the points into the slope formula: \( m = \frac{6 - 2}{5 - 7} \).

Step 3: Perform the Calculations

Calculate both the numerator and the denominator: \( 6 - 2 = 4 \) and \( 5 - 7 = -2 \).

Step 4: Simplify the Expression

Simplify the fraction \( \frac{4}{-2} \). Divide both the numerator and the denominator by 2, which gives \( -2 \).

Step 5: Conclusion

The slope of the line that passes through the points \((7, 2)\) and \((5, 6)\) is \(-2\).

Key Concepts

Coordinate Geometry

In the magical world of coordinate geometry, we plot points on a plane to better understand the relationships between them. Imagine a blank graph paper where each point is defined by a pair of numbers (x, y). These numbers are called coordinates. The first number, x, tells us how far to move horizontally from the origin (0,0), while y tells us how far to move vertically.
When we discuss the line connecting two points such as \((7,2)\)and \((5,6)\), we simply pinpoint these locations on our graph.

• The first point \((7,2)\) means that we need to go 7 units right and 2 units up from the origin.


• The second point \((5,6)\) directs us to move 5 units right and 6 units up.

The line through these points tells us how steeply the path moves upwards or downwards as we transition from one point to the other.By using coordinate geometry, we make sense of this movement and learn about different properties, such as the slope, of the line.

Slope Formula

Whenever you hear about a slope, think of it as a way to measure how slanted a line is. Is it going steep uphill or downhill, or is it level like the horizon? That's where the magical slope formula comes in handy: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).This formula is our trusty tool that helps in finding the slope.
The variables \( y_2 \) and \( y_1 \) in the formula represent the y-coordinates of the second and first point, respectively. Meanwhile, \( x_2 \) and \( x_1 \) are the x-coordinates. By plugging these values into the formula, we compute the difference between the y-values and divide it by the difference between the x-values.

• This quotient expresses the steepness between the two points.
• If the result is positive, the line rises as it moves from left to right, and if negative, as in our exercise, it falls.

The formula elegantly captures the essence of the line's inclination.

Mathematical Calculations

When it comes to numbers, small calculations have big implications. Using the slope formula, we substitute the coordinates we have. Looking at our points \((7,2)\)and \((5,6)\), let's dig into the math.
We start by plugging these into our formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

• Identify each point: \( (x_1, y_1) = (7, 2) \) and \( (x_2, y_2) = (5, 6) \).
• Find the change in y-coordinates: \( 6 - 2 = 4 \).
• Find the change in x-coordinates: \( 5 - 7 = -2 \).
• Divide these differences: \( m = \frac{4}{-2} \).

By simplifying \( \frac{4}{-2} \) to \(-2\), we discover the slope. A slope of \(-2\) means the line falls steeply. For every move of one unit to the right, the line drops down by 2 units. Notice how understanding each simple step carefully can illuminate exactly how the slope is calculated and what it represents!