Problem 46 - Elementary Algebra

Question

Find the slope, if it exists, of the line through the given pairs of points. $$ (5,2), \quad(6,3) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 1.

Step 1: Write the coordinates of the given points

First, write the coordinates of the given points as \((x_1, y_1) = (5,2)\) and \((x_2, y_2) = (6,3)\).

Step 2: Apply the slope formula

Next, apply the slope formula with the coordinates of the given points: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Step 3: Substitute the coordinates and calculate the slope

Substitute the coordinates of the given points into the slope formula and calculate the slope: $$ m = \frac{3 - 2}{6 - 5} = \frac{1}{1} = 1 $$

Step 4: State the slope of the line

The slope of the line through the given points \((5,2)\) and \((6,3)\) is \(1\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is an essential branch of mathematics that merges algebra with geometry. This allows us to use coordinates in a plane to solve geometric problems.
In coordinate geometry, each point on a plane is defined by an ordered pair

• The x-coordinate indicates the horizontal position (left-right).
• The y-coordinate shows the vertical position (up-down).

For example, the point (5,2) consists of: - An x-coordinate of 5, showing its position along the horizontal axis. - A y-coordinate of 2, placing it along the vertical axis.
By understanding where each point is, we're able to define straight lines, curves, and other geometric shapes, and perform calculations like finding the slope, distance, or equation of a line.

Slope Formula

The slope of a line represents its steepness and direction. It's a crucial concept in coordinate geometry and is calculated using the slope formula:
The slope formula is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:

• \(m\) is the slope,
• \((x_1, y_1)\) are the coordinates of the first point,
• \((x_2, y_2)\) are the coordinates of the second point.

This formula essentially measures the change in the vertical direction (rise) over the change in the horizontal direction (run). For instance, for the points (5,2) and (6,3), the slope is calculated as follows:- Calculate the difference in the y-coordinates: \( 3 - 2 = 1 \) (the "rise").- Calculate the difference in the x-coordinates: \( 6 - 5 = 1 \) (the "run").
The slope ends up being \( \frac{1}{1} = 1 \), indicating a positive slope with an upward direction from left to right.

Linear Equations

Linear equations involve expressions that represent lines on a graph. They can be expressed in different forms, but the most common ones are in slope-intercept form and point-slope form.

The slope-intercept form is:\[ y = mx + b \]Where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

For example, if a line has a slope of 1 and crosses the y-axis at 0, it can be written as \(y = x\).

The point-slope form is quite useful if you know the slope of the line and a point on it. This form is:\[ y - y_1 = m(x - x_1) \]Using point (5,2) and a slope of 1, the equation becomes \(y - 2 = 1(x - 5)\), displaying the line starting at (5,2) with a consistent slope.
Understanding these forms allows you to graph lines and solve problems involving intersections and parallelism in coordinate geometry.