Problem 29 - Algebra for College Students

Question

Find \(x\) if the line through \((-2,4)\) and \((x, 6)\) has a slope of \(\frac{2}{9}\).

Step-by-Step Solution

Answer

The value of \(x\) is 7.

Step 1: Understand the Slope Formula

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. In this problem, you are given a slope, m = \(\frac{2}{9}\), and the points (-2, 4) and (x, 6), where x is unknown.

Step 2: Apply Given Values to the Slope Formula

Using the slope formula with the given points (-2, 4) and (x, 6), substitute the values: \[ \frac{6 - 4}{x - (-2)} = \frac{2}{9} \]. Simplify to get: \[ \frac{2}{x + 2} = \frac{2}{9} \].

Step 3: Solve for x

Since the fractions are equal, set the denominators equal: \(x + 2 = 9\). Solve for x by subtracting 2 from both sides: \[ x = 9 - 2 \]. Thus, \[ x = 7 \].

Key Concepts

Slope Formula

The slope formula is essential for understanding linear relationships in algebra. It helps us determine how steep or flat a line is between two specific points. When we talk about slope in algebra, we're referring to the change in the vertical direction (up-down) compared to the change in the horizontal direction (left-right). This is often represented as the ratio \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on a line.

• Numerator: The "rise" or vertical change between the two points.
• Denominator: The "run" or horizontal change between the two points.

Understanding the slope concept is like understanding the angle or inclination of something. A larger slope means a steeper line, while a smaller slope indicates a flatter line. Positive slope values suggest the line goes upwards as it moves from left to right, while negative slopes suggest a downward trend. Mastering the slope formula gives you a strong foundation for tackling various linear equations.

Linear Equations

Linear equations form the basis of algebra and involve expressions of the first degree, meaning the variable is not raised to any power other than one. These equations typically have the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the place where the line crosses the y-axis. However, linear equations can also be represented in a point-slope form or standard form.

• Point-Slope Form: Utilizes a given point and slope to model the line, expressed as \( y - y_1 = m(x - x_1) \).
• Standard Form: Typically written as \( Ax + By = C \), focusing more on the coefficients and constant terms.

Grasping linear equations is about understanding how straight lines can be described using algebraic expressions. They are simple yet powerful tools to model and solve real-world problems that can be visualized on a graph with a straight line.

Solving for Variables

Solving for variables is a crucial skill in algebra. It involves figuring out the value of unknowns that make an equation true. In this context, the variable \( x \) needs to be isolated to find its value. Let's break this down using our example:

• When given the slope \( \frac{2}{9} \) and points \((-2, 4)\) and \((x, 6)\), we formulated the equation \( \frac{2}{x+2} = \frac{2}{9} \).
• Since the numerators are the same, we focus on equaling the denominators so \( x + 2 = 9 \).
• Then, simply subtract \( 2 \) from both sides to solve for \( x \), resulting in \( x = 7 \).

The process of solving equations often requires manipulation, like addition or subtraction, to isolate the variable. It lays the foundation for more complex problem-solving and helps reinforce logic and reasoning skills in mathematics.