Problem 18 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ (1824,108),(1900,380) $$

Step-by-Step Solution

Answer

The slope is \(\frac{68}{19}\).

Step 1: Understand the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We'll use this formula to find the slope.

Step 2: Identify Coordinates

We will assign our coordinates: \((x_1, y_1) = (1824, 108)\) and \((x_2, y_2) = (1900, 380)\).

Step 3: Substitute into Slope Formula

Substitute the coordinates into the slope formula: \( m = \frac{380 - 108}{1900 - 1824} \).

Step 4: Simplify the Subtraction in the Numerator

Calculate \(380 - 108\) which equals \(272\).

Step 5: Simplify the Subtraction in the Denominator

Calculate \(1900 - 1824\) which equals \(76\).

Step 6: Calculate the Slope

Divide the difference in the \(y\)-coordinates by the difference in the \(x\)-coordinates to find the slope: \( m = \frac{272}{76} \).

Step 7: Simplify the Fraction

Reduce the fraction \( \frac{272}{76} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So, \( m = \frac{68}{19} \).

Key Concepts

Coordinate geometry

Coordinate Geometry is a branch of mathematics that ties algebra and geometry together using a coordinate plane. This is a system that uses two number lines, perpendicular to each other, creating a grid to visualize numerical relationships. The horizontal line is called the x-axis, while the vertical line is the y-axis.

The intersection of these axes, known as the origin, has coordinates (0,0). Each point on this grid has an x-coordinate and a y-coordinate, which helps us determine the point's position. For example, in our coordinate pair

• (1824, 108), the x-coordinate is 1824 and the y-coordinate is 108.


• Similarly, for (1900, 380), 1900 is the x-coordinate and 380 is the y-coordinate.

Understanding these coordinates is the basis to solve problems involving lines, such as calculating the slope between these two points through the slope formula.

Linear equations

A linear equation is an algebraic equation that forms a straight line when graphed on a coordinate plane. It generally takes the form

• \(y = mx + b\)

where \(m\) is the slope of the line and \(b\) is the y-intercept. The y-intercept is the point where the line crosses the y-axis (where \(x = 0\)). This equation is crucial because it helps us understand how the value of y changes with x, fully defining the behavior of the line.

In this particular problem, though an explicit equation isn't given, finding the slope

• (which is \( \frac{68}{19} \), in simplified form, from the original problem)

provides key information about the line passing through the given points. Once the slope is known, other aspects of the line, like intercepts, can be determined if additional points or information are provided.

Algebraic expressions

Algebraic expressions involve numbers, variables, and operational symbols. They allow us to solve for unknowns and express mathematical relationships efficiently. For example, the slope of a line is calculated using the algebraic formula

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this expression, \(m\) represents the slope, and it involves the variables \(x_1, x_2, y_1, \) and \(y_2\), which represent the coordinates of the points.

During the problem-solving process, algebraic manipulation is critical. Subtraction was used to find the differences in y (\(380-108 = 272\)) and x (\(1900-1824 = 76\)).
Finally, division simplified the equation and determined the slope \(\frac{272}{76}\), which was further reduced to \(\frac{68}{19}\). Working with algebraic expressions like this is fundamental in many areas of mathematics, from solving equations to performing more complex operations.