Problem 68 - Algebra 1: Concepts and Skills

Question

Find the slope of the line that passes through the points. \((0,3)\) and \((2,1)\)

Step-by-Step Solution

Answer

The slope of the line that passes through the points \((0,3)\) and \((2,1)\) is -1.

Step 1 - Identify the coordinates of the points

The coordinates for the first point are \((x_1, y_1) = (0, 3)\). The coordinates for the second point are \((x_2, y_2) = (2, 1)\).

Step 2 - Substitute into the formula

Next, substitute these values into the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 3}{2 - 0}\).

Step 3 - Simplify the expression

Simplify the expression above: \(m = \frac{-2}{2} = -1\)

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that combines algebra with geometry to describe positions and relationships between points, lines, and figures using a coordinate plane. Imagine drawing a picture on a graph. Each point in your picture is described by a pair of numbers called coordinates.
In a 2D coordinate plane, any location is given by two numbers in parentheses like this: \((x, y)\). The first number \(x\) is how far along you go (left to right), and the second number \(y\) is how far up or down you move.
In our task, the points \((0, 3)\) and \((2, 1)\) are each coordinates that are used to find the slope of a line. Understanding coordinate geometry is crucial because it helps us depict spatial relationships concretely through numbers and equations. This helps in various applications like mapping, navigation, and even computer graphics.

Slope Formula

The Slope Formula is a handy tool in coordinate geometry that calculates the steepness or incline of a line passing through two points. The slope is represented by the letter \(m\) and is key to understanding how a line behaves on a graph.
To find the slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\), use this formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Think of this as finding the "rise" over the "run," which are changes in vertical and horizontal positions, respectively.

• "Rise" is the difference in the \(y\)-coordinates (vertical change).
• "Run" is the difference in the \(x\)-coordinates (horizontal change).

Substituting our points \((0, 3)\) and \((2, 1)\), we get \(m = \frac{1 - 3}{2 - 0} = \frac{-2}{2} = -1\).
This negative slope means the line goes down as it moves from left to right.

Linear Equations

Linear Equations form the backbone of algebra involving lines. They describe a straight line in terms of numbers and variables. The basic form of a linear equation is \(y = mx + c\), where:

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

From the information given, knowing the slope \(m = -1\) and a point \((0, 3)\) the line passes through, we can write the equation for this line. Because the point \((0, 3)\) lies on the \(y\)-axis, \(3\) is the \(y\)-intercept \(c\).
Therefore, the linear equation for the line is \(y = -x + 3\).
This equation tells us how to compute \(y\) for any \(x\) along that line and helps predict other points on the line. Understanding linear equations is essential because they simplify problem-solving across sciences and financial calculations.