Problem 37 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. $$(24.3,11.9),(3.57,8.4)$$

Step-by-Step Solution

Answer

The slope is approximately 0.169.

Step 1: Identify the coordinates

The given points are \(24.3, 11.9\) and \(3.57, 8.4\). Label them as \(x_1, y_1\) and \(x_2, y_2\) respectively.\(x_1 = 24.3, y_1 = 11.9\)\(x_2 = 3.57, y_2 = 8.4\)

Step 2: Apply the slope formula

The formula to find the slope \(m\) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Input the coordinates into the formula

Substitute the values into the formula: \[ m = \frac{8.4 - 11.9}{3.57 - 24.3} \]

Step 4: Simplify the numerator

Calculate the difference between \(y_2\) and \(y_1\): \[ 8.4 - 11.9 = -3.5 \]

Step 5: Simplify the denominator

Calculate the difference between \(x_2\) and \(x_1\): \[ 3.57 - 24.3 = -20.73 \]

Step 6: Divide the results

Divide the simplified numerator by the simplified denominator: \[ m = \frac{-3.5}{-20.73} \]

Step 7: Simplify the fraction

Simplify the fraction to find the final slope value: \[ m \approx 0.169 \]

Key Concepts

slope formula

The slope formula is a fundamental concept in mathematics, especially in algebra and linear equations. It helps us determine the steepness of a line between two points on a graph. The formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This means that 'm' (the slope) is found by dividing the difference in the y-coordinates by the difference in the x-coordinates. Remember:

• \(y_2\) and \(y_1\) are the y-values of two different points.
• \(x_2\) and \(x_1\) are the x-values of the same points.

By using this formula, you can calculate how much the line rises or falls as it moves from left to right.

coordinates

Coordinates are pairs of numbers that locate points on a graph. They are always in the form \((x, y)\), where 'x' represents the horizontal position and 'y' the vertical position.
In our example, the coordinates are \((24.3, 11.9)\) and \((3.57, 8.4)\). Here:

• For the first point, \(x_1 = 24.3\) and \(y_1 = 11.9\).
• For the second point, \(x_2 = 3.57\) and \(y_2 = 8.4\).

Coordinates are essential for plotting points and understanding the position of a line on a graph.

linear equations

Linear equations represent straight lines in a graph. They often come in the form \(y = mx + b\), where:

• 'm' is the slope.
• 'b' is the y-intercept, or the point where the line crosses the y-axis.

To understand a linear equation, you need to know the slope and at least one point on the line. Once you have the slope, you can use it to find more points that lie on the line.

algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's about finding the unknowns in equations. When you solve for the slope of a line, you're applying the principles of algebra.
In the given problem:

• You identify the coordinates of points.
• Apply the slope formula.
• Simplify the numbers step by step.

These processes are all about using algebraic methods to find an answer. Algebra helps you break down complex problems into manageable steps.