Problem 29 - Intermediate Algebra

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: Identify the formula for slope

The slope formula, which calculates the steepness of a line through two points, is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, the slope \( m \) is \( \frac{2}{9} \), and the points are \((-2, 4)\) and \((x, 6)\).

Step 2: Substitute the known values into the slope formula

Substitute \( y_2 = 6 \), \( y_1 = 4 \), \( x_1 = -2 \), and \( x_2 = x \) into the slope formula. This gives: \( \frac{2}{9} = \frac{6 - 4}{x - (-2)} \).

Step 3: Simplify the slope equation

Simplify the equation \( \frac{6-4}{x - (-2)} \) to \( \frac{2}{x + 2} \), resulting in the equation \( \frac{2}{9} = \frac{2}{x + 2} \).

Step 4: Solve for x

To solve for \( x \), set the equation \( \frac{2}{9} = \frac{2}{x+2} \). Cross-multiply to get \( 2(x+2) = 18 \). Simplify to find \( 2x + 4 = 18 \).

Step 5: Isolate x

Subtract 4 from both sides to get \( 2x = 14 \). Then, divide both sides by 2 to solve for \( x \), giving \( x = 7 \).

Key Concepts

Linear Equations

In mathematics, a linear equation is an equation that models a straight line on a graph. These equations often appear in the form \(y = mx + b\), where:

• \(m\) is the slope, representing the steepness or incline of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

Understanding linear equations allows you to predict and measure changes from one variable to another—a skill that is vital in many fields such as economics, engineering, and science.
For the exercise, we used a specialized version of linear equations, focusing on the calculation of slope. The slope provides insight into the growth rate or decline between two points on a line, a critical component in determining the value of an unknown \(x\).

Algebra

Algebra forms the foundation of modern mathematics, using symbols and letters to represent numbers. This field of math allows equations to be solved, connecting variables in ways that arithmetic alone cannot achieve.
In the current exercise, algebraic techniques were crucial for isolating and determining the variable \(x\). Let's break down how it worked:

• First, the slope formula was set as an equation: \(\frac{2}{9} = \frac{2}{x+2}\).
• Using algebraic manipulation, we cross-multiplied to eliminate the fractions, resulting in: \(2(x+2) = 18\).
• Next, we expanded the equation to get \(2x + 4 = 18\), then isolated \(x\) by performing inverse operations, eventually solving for \(x = 7\).

These steps demonstrate algebra's power to rearrange terms and solve for unknowns, a core skill in mathematics.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry using a coordinate system. This allows for the precise plotting and analysis of geometric figures on a plane using numerical coordinates.
The exercise beautifully illustrates coordinate geometry concepts by using points \((-2, 4)\) and \((x, 6)\). Here’s how it worked in finding the slope:

• Point \((-2, 4)\) represents one point on the line, marking the beginning of our line segment.
• Point \((x, 6)\) represents the other end, where \(x\) is the unknown variable to find.
• Both points were inserted into the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), enabling us to solve for \(x\).

Through coordinate geometry, seemingly abstract numerical relationships become visual and easier to understand—offering a solid framework for problem-solving.