Problem 29 - Beginning and Intermediate Algebra

Question

Use the slope formula to find the slope of the line containing each pair of points. $$\left(\frac{3}{2},-1\right) \text { and }\left(-\frac{5}{2}, 7\right)$$

Step-by-Step Solution

Answer

The slope of the line containing the points \(\left(\frac{3}{2},-1\right)\) and \(\left(-\frac{5}{2}, 7\right)\) is \(m = -4\).

Step 1: Identify the coordinates of the points

The given points are: Point 1: \(\left(\frac{3}{2},-1\right)\) with coordinates: \(x_1 = \frac{3}{2}\), \(y_1 = -1\) Point 2: \(\left(-\frac{5}{2}, 7\right)\) with coordinates: \(x_2 = -\frac{5}{2}\), \(y_2 = 7\)

Step 2: Use the slope formula

Plug in the coordinates of the points into the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - (-1)}{-\frac{5}{2} - \frac{3}{2}}$$

Step 3: Simplify the expression

Add the numerators and subtract the denominators: $$m = \frac{7 + 1}{-\frac{5 - 3}{2}} = \frac{8}{-2}$$ Now, we will divide the numerator by the denominator: $$m = -4$$

Step 4: State the final answer

So, the slope of the line containing the points \(\left(\frac{3}{2},-1\right)\) and \(\left(-\frac{5}{2}, 7\right)\) is \(m = -4\).

Key Concepts

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system. It's a crucial area in mathematics that combines algebra and geometry, enabling the description of shapes in a plane using numbers. This method uses a pair of numbers, called coordinates, to specify a point's position on a graph.

With coordinate geometry, you can easily map and visualize how shapes and lines interact. Points on a plane are usually represented as \((x, y)\), where \(x\) is the horizontal direction and \(y\) is the vertical direction. This approach is foundational for solving geometric problems algebraically.

Slope Formula

The slope formula is a mathematical tool used to find the steepness or incline of a line between two points. The formula is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

• The numerator \(y_2 - y_1\) represents the change in the \(y\)-values, often referred to as “rise.”
• The denominator \(x_2 - x_1\) represents the change in the \(x\)-values, or “run.”

A positive slope indicates a line going upward, while a negative slope shows a line going downward. If the slope is zero, the line is horizontal.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They are used to represent real-world quantities analytically, providing a means to solve equations and understand relationships between variables.

In the slope problem, algebraic expressions help simplify and solve complex numerical relationships. You might encounter expressions like \(\frac{7 + 1}{-\frac{5 - 3}{2}}\), where simplification yields the slope of the line. Understanding how to manipulate these expressions is key to gaining insights into various mathematical problems.

Line Equations

Line equations represent lines on a graph and describe their properties mathematically. The most common form of a line equation is the slope-intercept form, given by:
\[ y = mx + b \]where \(m\) is the slope and \(b\) is the y-intercept.

This equation helps graph lines easily by identifying the slope and the point where the line intersects the \(y\)-axis. Using the information about the slope between two points, you can often find the equation of the line that connects them.

Understanding line equations is crucial for analyzing linear relationships and graphing them effectively.