Problem 54 - Algebra 2

Question

Determine the value of \(r\) so that a line through the points with the given coordinates has the given slope. (Lesson \(2-3 )\) $$ (r, 2),(4,-6) ; \text { slope }=-\frac{8}{3} $$

Step-by-Step Solution

Answer

The value of \(r\) is 1.

Step 1: Understanding the slope formula

The slope, denoted as \( m \), of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In this problem, the given slope \( m \) is \(-\frac{8}{3}\), and the coordinates of the points are \((r, 2)\) and \((4, -6)\).

Step 2: Substitute coordinates in the slope formula

Substitute the given coordinates and the slope into the formula: \[-\frac{8}{3} = \frac{-6 - 2}{4 - r} \] This equation sets up a proportion involving the slope.

Step 3: Solving the equation for \(r\)

First, simplify the numerator in the equation: \(-6 - 2 = -8\). Thus, the equation becomes: \[-\frac{8}{3} = \frac{-8}{4 - r} \] To solve for \(r\), cross-multiply to eliminate the fractions:\[-8(4 - r) = -8 \times 3\] Which simplifies to: \[-8(4 - r) = -24\]

Step 4: Simplifying the equation

Distribute the \(-8\) on the left side the equation: \[-8(4) + 8r = -24\] Which simplifies to: \[-32 + 8r = -24\].Next, add \(32\) to both sides of the equation: \[8r = -24 + 32\] This simplifies to: \[8r = 8\]

Step 5: Find the value of \(r\)

To solve for \(r\), divide both sides by \(8\): \[r = \frac{8}{8}\] This simplifies to \(r = 1\).

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in algebra and coordinate geometry. It measures the steepness or inclination of a line and is represented by the symbol \( m \). The slope is calculated using the formula:

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

where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
The numerator \( y_2 - y_1 \) represents the change in the vertical direction, often called "rise," and the denominator \( x_2 - x_1 \) is the change in the horizontal direction, also called "run."
When the slope is negative, such as in this exercise \(-\frac{8}{3}\), the line decreases as it moves from left to right.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. It connects algebra and geometry by describing geometric shapes and properties through algebraic equations.
In this exercise, coordinate geometry is used to determine the relationship between points and the slope of the line they form:

• You start with two points, \((r, 2)\) and \((4, -6)\), which are represented on the coordinate plane.
• By applying the formula for slope, you incorporate the values from these coordinates to form an algebraic equation.
• The equation is based on the displacement between these two points, which can be visually interpreted as the slope of the line connecting them.

Coordinate geometry not only allows solving algebraic problems related to shapes, sizes, and positions but also adds depth for graphical interpretation and analysis.

Solving Equations

Solving equations is a critical skill in algebra. It involves finding the value of an unknown variable that satisfies a given equation.
Here is how you can solve an equation systematically:

• First, replace the points and slope into the slope formula to form an equation: \[-\frac{8}{3} = \frac{-8}{4 - r}\]
• Simplify both sides of the equation, if needed. In this exercise, after simplifying the numerator, you perform cross-multiplication to get rid of the fraction: \[-8(4 - r) = -24\]
• Distribute and simplify to isolate the term with the variable: \[-32 + 8r = -24\]
• Finally, solve for \( r \) by performing arithmetic operations to isolate the variable on one side of the equation: \[8r = 8\]
• Conclude by dividing to find \( r \): \[r = 1\]

Mastering these steps provides a robust foundation for tackling more complex algebraic problems in algebra 2 and beyond.