Problem 23 - Elementary Algebra

Question

Find \(x\) if the line through the points \((-2,-4)\) and \((x, 2)\) has a slope of \(-\frac{3}{2}\).

Step-by-Step Solution

Answer

The value of \(x\) is -6.

Step 1: Understand the Slope Formula

The formula for calculating the slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, the slope is \(-\frac{3}{2}\), and the points are \((-2,-4)\) and \((x, 2)\).

Step 2: Plug Points and Slope into the Formula

Substitute the given points and the slope into the slope formula:\[ -\frac{3}{2} = \frac{2 - (-4)}{x - (-2)} \]This simplifies to:\[ -\frac{3}{2} = \frac{2 + 4}{x + 2} \]

Step 3: Simplify the Slope Equation

Further simplify the equation:\[ -\frac{3}{2} = \frac{6}{x + 2} \]

Step 4: Cross-Multiply to Solve for \(x\)

Cross-multiply to clear the fraction:\[ -3(x + 2) = 12 \]Distribute the \(-3\):\[ -3x - 6 = 12 \]

Step 5: Isolate \(x\)

Add 6 to both sides to eliminate the constant term:\[ -3x = 18 \]Divide both sides by \(-3\) to solve for \(x\):\[ x = -6 \]

Key Concepts

Slope Formula

The slope formula is a fundamental component in algebra for describing the steepness, or incline, of a line on a coordinate plane. It is usually represented by the letter "\(m\)" and can be determined using the coordinates of two distinct points on a line. If you have two points: \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This expression tells you how much the \(y\)-value (vertical change) changes between two points for every one-unit change in the \(x\)-value (horizontal change).

• A positive slope indicates an upward trend from left to right.
• A negative slope suggests a downward trend.
• A zero slope means the line is horizontal.
• A slope that is undefined corresponds to a vertical line.

Understanding how to apply the slope formula is crucial when analyzing line graphs or when you need to derive equations of lines.

Linear Equations

Linear equations are equations where the variable occurs to the power of one and graphs into a straight line on a coordinate plane. These equations typically look like \(y = mx + b\), where \(m\) and \(b\) are constants:

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

Linear equations can represent a vast number of real-world situations, where the relationship between two quantities is constant. Solving linear equations might involve solving for \(x\) or \(y\), typically by isolating the variable on one side of the equation. This can involve simple arithmetic operations like addition, subtraction, multiplication, or division. In more complex equations, you may see techniques such as substitution or elimination, especially in systems of linear equations. The versatility of linear equations makes them a centerpiece of algebra and essential for understanding more complex mathematical concepts.

Cross-Multiplication

Cross-multiplication is a straightforward technique often used to solve equations involving fractions. In essence, when you have a fraction equal to another fraction, you can "cross" multiply the diagonals to eliminate the denominators. Through cross-multiplication, the equation \(\frac{a}{b} = \frac{c}{d}\) turns into:\[ a \cdot d = b \cdot c \] This method simplifies the process of solving for unknown variables, especially in linear relationships where fractions can be present, like slope calculations. In the context of the given problem, this method helps to shift away from fractions, making it easier to focus on isolating and solving for the variable. Cross-multiplication is not only beneficial for simplification purposes but is also a useful algebraic tool within proportional relationships. It provides a quick way to find unknowns and is particularly helpful in solving many different algebraic equations efficiently.