Problem 55 - Algebra 2

Question

Determine the value of \(r\) so that a line through the points with the given coordinates has the given slope. (Lesson \(2-3 )\) $$ (r, 6),(8,4) ; \text { slope }=\frac{1}{2} $$

Step-by-Step Solution

Answer

The value of \(r\) is 9.

Step 1: Recall the Slope Formula

The formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this problem, we are given that the slope \(m\) is \(\frac{1}{2}\).

Step 2: Set Up the Equation

Substitute the coordinates \((r, 6)\) and \((8, 4)\) into the slope formula. This gives the equation:\[ \frac{4 - 6}{8 - r} = \frac{1}{2} \]

Step 3: Simplify the Left Side

Calculate the difference in the y-coordinates and simplify:\[ \frac{4 - 6}{8 - r} = \frac{-2}{8 - r} \]This results in:\[ \frac{-2}{8 - r} = \frac{1}{2} \]

Step 4: Solve for \(r\)

Cross multiply to solve for \(r\):\[ -2 \times (8 - r) = 1 \times 2 \]Simplify the equation:\[ -16 + 2r = 2 \]Add 16 to both sides:\[ 2r = 18 \]Divide by 2:\[ r = 9 \]

Key Concepts

Slope Formula

The slope formula is a fundamental concept in algebra and coordinate geometry. It helps us understand the steepness of a line by calculating the ratio of the change in the vertical direction to the change in the horizontal direction between two points. Given two coordinates

• \((x_1, y_1)\)
• \((x_2, y_2)\)

the formula can be expressed as follows:\[ m = \frac{y_2 - y_1}{x_2 - x_1}.\]This expression tells us how much "rise," or change in y, occurs per "run," or change in x. The slope is often denoted by the letter \(m\) and can be a positive, a negative value, zero, or undefined, depending on the direction and line's inclination.
Here are some key points to remember:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope means the line is horizontal.
• An undefined slope means the line is vertical.

By using the slope formula, you can determine the direction and steepness of a line connecting two points on a coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate plane to represent geometric shapes. The coordinate plane consists of two perpendicular lines called axes:

• The horizontal line is the x-axis.
• The vertical line is the y-axis.

Coordinates are always written in pairs

• \((x, y)\)

where \(x\) is the position along the x-axis, and \(y\) is the position along the y-axis. Points are plotted on this plane using these coordinates. Coordinate geometry lets you perform geometric transformations and calculations using algebra. It offers tools like the distance formula, midpoint formula, and the slope formula to analyze the relationships between points. When working with line equations or interpreting linear graphs, coordinate geometry provides a clear framework for deriving important properties and insights. This technique is essential for solving problems involving the position and shape of figures in a plane.

Solving Equations

Solving equations is a crucial skill in algebra. It involves finding the unknowns that satisfy given equations. In the exercise, solving for \(r\) required simplifying and rearranging an equation derived from the slope formula.Here’s a recap of the process:1. Start with the equation from the slope calculation: \[\frac{-2}{8 - r} = \frac{1}{2} \]2. Use cross-multiplication to eliminate fractions: \[-2 \times (8 - r) = 1 \times 2 \]3. Simplify and solve: - Expand the left side: \(-16 + 2r = 2\) - Add 16 to both sides to isolate terms with \(r\): \[2r = 18\] - Divide by 2 to solve for \(r\): \[r = 9\]These steps demonstrate how algebra is used to manipulate equations, ensuring that both sides remain equal as changes are made. Each operation is designed to unravel the unknown, making algebra a powerful tool for handling different mathematical problems.