Problem 21 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((-3,-1),(4,3)\)a

Step-by-Step Solution

Answer

The slope is \( \frac{4}{7} \).

Step 1: Identify the coordinates

Identify the given points on the line: Point 1: (x_1, y_1) = (-3, -1) Point 2: (x_2, y_2) = (4, 3)

Step 2: Write the slope formula

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

Step 3: Substitute the coordinates into the slope formula

Substitute the coordinates of the given points into the slope formula: m = \( \frac{3 - (-1)}{4 - (-3)} \)

Step 4: Simplify the expression

Simplify the numerator and the denominator: m = \( \frac{3 + 1}{4 + 3} \) = \( \frac{4}{7} \)

Key Concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we describe geometric figures using a coordinate system. In this system, each point is defined by an ordered pair of numbers (x, y) which represent its location on the plane. This makes it possible to calculate various properties of geometric figures, such as the slope of a line, distance between points, and equations of shapes. By using ordered pairs, we can conveniently describe and solve problems involving points, lines, and shapes in a two-dimensional space.

slope formula

The slope of a line is a measure of how steep the line is. It’s a key property in coordinate geometry. The slope is represented by the letter 'm' and is calculated using the slope formula. This formula uses two points on the line, given as \( (x_1, y_1) \) and \( (x_2, y_2) \). The formula is:

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

Here’s what each term means:

• \( y_2 - y_1 \): This is the difference between the y-coordinates of the two points, also called the rise.
• \( x_2 - x_1 \): This is the difference between the x-coordinates of the two points, also called the run.

By applying this formula, we can determine whether a line is rising, falling, or horizontal. For example, in our exercise, substituting \( (x_1, y_1) = (-3, -1) \) and \( (x_2, y_2) = (4, 3) \), we get:

m = \( \frac{3 - (-1)}{4 - (-3)} \)

Simplifying this, we find m = \( \frac{4}{7} \). This means the line rises moderately as it moves from left to right.

linear equations

Linear equations represent straight lines in coordinate geometry. They generally take the form:

y = mx + b

where:

• y is the dependent variable
• m is the slope of the line
• x is the independent variable
• b is the y-intercept, which is the point where the line crosses the y-axis.

To find the equation of a line when you know its slope and a point on the line, you can rearrange the formula.
For instance, using the slope m = \( \frac{4}{7} \) from our solution and the point (4, 3):
\( y - y_1 = m(x - x_1) \)

Substituting in the values:
\( y - 3 = \frac{4}{7}(x - 4) \)

Solving for y gives the linear equation in slope-intercept form. Linear equations help us graph lines and understand relationships between variables.