Problem 44 - Algebra 2

Question

Determine the value of \(r\) so that the line through \((6, r)\) and \((9,2)\) has slope \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The value of \(r\) is 1.

Step 1: Formula for Slope

Recall the formula for the slope of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(m\) is the slope of the line.

Step 2: Substituting Known Values

Substitute the given points \((6, r)\) and \((9, 2)\) into the slope formula, and set the slope equal to \(\frac{1}{3}\): \[ \frac{2 - r}{9 - 6} = \frac{1}{3} \]

Step 3: Simplify the Equation

Simplify the denominator: \[ \frac{2 - r}{3} = \frac{1}{3} \]

Step 4: Solve for \(r\)

Since the denominators are the same, equate the numerators: \[ 2 - r = 1 \]Now, solve for \(r\) by rearranging the equation: \[ 2 - 1 = r \] \[ r = 1 \]

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in algebra, especially when dealing with linear equations. The slope indicates the steepness and the direction of a line on a graph. It is defined as the ratio of the line's vertical change, often called the "rise," to its horizontal change, known as the "run." For any two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula for the slope \( m \) is given by:

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

This formula helps determine how slanted the line is. A positive slope indicates that the line rises from left to right, while a negative slope means it falls. A slope of zero means the line is horizontal and a vertical line has an undefined slope. Understanding these concepts will allow you to interpret the behavior of the graph efficiently.

Linear Equations

Linear equations form the foundation of algebra. They represent equations of the first degree, which means their highest power is 1. A linear equation in two variables, \(x \) and \(y\), can be represented in the standard form:

• Standard form: \(Ax + By = C \)
• Slope-intercept form: \(y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.

When plotting a linear equation, it forms a straight line on a coordinate plane. The slope and the intercepts give critical information about the line's orientation and position. The understanding of linear equations helps solve systems of equations and analyze real-world situations like calculating rates or trends.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves understanding geometric shapes in a numerical manner through the use of coordinates. By using the Cartesian coordinate plane, we can place points, plot lines, and analyze geometric properties using algebraic equations. The Cartesian plane consists of two axes intersecting perpendicularly: the x-axis (horizontal) and the y-axis (vertical).
Points are represented as ordered pairs \( (x, y) \), where \(x\) denotes the horizontal position and \(y\) represents the vertical position. Calculating the distance between points, finding midpoints, and determining areas are common tasks facilitated by coordinate geometry.
Through this branch of mathematics, we can mesh algebra with geometry, which provides powerful tools for solving problems involving shapes and figures. It bridges abstract algebraic concepts with real, visual representation, making complex problems more approachable.