Problem 25 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((a, b)\) and \((2 a, 2 b), a \neq 0\)

Step-by-Step Solution

Answer

The slope is \ \frac{b}{a} \.

Step 1: Identify the General Slope Formula

The slope formula for a line passing through two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is given by \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \).

Step 2: Assign the Points to Variables

Assign \((a, b)\) to \((x_{1}, y_{1})\) and \((2a, 2b)\) to \((x_{2}, y_{2})\). So we have \((x_{1}, y_{1}) = (a, b)\) and \((x_{2}, y_{2}) = (2a, 2b)\).

Step 3: Substitute the Points into the Slope Formula

Substitute the coordinates into the slope formula: \( m = \frac{2b - b}{2a - a} = \frac{b}{a} \).

Step 4: Simplify the Slope Expression

Simplify the fraction obtained: \( m = \frac{b}{a} \).

Key Concepts

Slope Formula

The slope formula is a key part of understanding linear equations and coordinate geometry. It helps determine the steepness or incline of a line. The slope of a line passing through two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) can be found using the formula: \(\text{m} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). This formula calculates how much a line rises or falls for each unit it moves horizontally. Remember, the numerator represents the change in the y-values (rise) and the denominator represents the change in x-values (run). When the slope is positive, the line rises to the right. When negative, it falls to the right. Zero slope indicates a horizontal line, while undefined slope (division by zero) indicates a vertical line.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that uses algebra to describe geometric principles. It involves the study of geometric figures using a coordinate system, which defines every point by a pair of numerical coordinates. This system allows us to analyze geometric shapes in two dimensions using algebraic equations. Points are defined by coordinates (x, y), reflecting their position in the plane. Lines are defined by linear equations. By applying the slope formula in coordinate geometry, we can easily calculate and graph lines. Understanding the coordinates gives us a clear picture of how these points and lines interact in space, making it simpler to solve various geometric problems.

Linear Equations

Linear equations are foundational in algebra and relate directly to lines in the coordinate plane. A linear equation in two variables usually takes the form \(y = mx + b\), where \(\text{m}\) is the slope and \(\text{b}\) is the y-intercept. The slope-intercept form \(y = mx + b\) tells us how fast y changes with respect to x (the slope \(m\)), and where the line crosses the y-axis (the intercept \(b\)). Another common form is the point-slope form \(y - y_{1} = m(x - x_{1})\), which uses a known point on the line to solve for other points. Linear equations can represent real-world relationships, making them crucial for practical applications such as physics, economics, and more. By mastering the manipulation of these equations, students can solve complex problems involving rates of change, intersections, and parallel/perpendicular lines.