Problem 17 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(-1,10),(-9,2)$$

Step-by-Step Solution

Answer

The slope is 1.

Step 1: Understand Slope Formula

The formula to find the slope \( m \) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where \( y_2 \) and \( y_1 \) are the \( y \)-coordinates of the points and \( x_2 \) and \( x_1 \) are the \( x \)-coordinates of the points.

Step 2: Substitute Coordinates into the Formula

Substitute the given points \((-1, 10)\) and \((-9, 2)\) into the slope formula.Let \((x_1, y_1) = (-1, 10)\) and \((x_2, y_2) = (-9, 2)\).Substitute these into the slope formula:\[ m = \frac{2 - 10}{-9 + 1} \]

Step 3: Simplify the Expression

Simplify the expression in the numerator and the denominator.Numerator: \(2 - 10 = -8\)Denominator: \(-9 + 1 = -8\)Thus, the expression becomes:\[ m = \frac{-8}{-8} \]

Step 4: Calculate the Slope Value

Now, calculate the value of the fraction:\[ \frac{-8}{-8} = 1 \]This gives us the slope \( m = 1 \).

Key Concepts

Algebra: Understanding the Slope

In algebra, the slope of a line is a crucial concept that determines the steepness or incline of the line. The slope can be calculated using the slope formula, which in essence, measures the change in vertical distance over the change in horizontal distance between two points. This calculation is expressed as:

• Numerator: The difference in the y-values (\( y_2 - y_1 \)), which indicates how much the line rises or falls.
• Denominator: The difference in the x-values (\( x_2 - x_1 \)), showing how much the line runs horizontally.

Crucially, the order of subtraction is essential; swapping these values around will alter the sign and consequently the slope's direction. If results in a positive value, the line ascends from left to right; if negative, it descends. This foundational algebraic tool enables you to interpret and analyze linear relationships in coordinate geometry effectively.

Coordinate Geometry: Points and Slopes

In coordinate geometry, every point on a plane can be denoted using a pair of coordinates \((x, y)\). When you have a pair of such points, like \((-1, 10)\) and \((-9, 2)\), you can calculate the slope of the line connecting them. Here’s how:

• Plot both points on a coordinate plane.
• Draw a line joining these points.
• Use the slope formula to find how steep the line is.

Understanding how these coordinates relate to each other through the slope formula helps us appreciate the direct impact of changes in \(x\) or \(y\) values. When simplifying the expression \( m = \frac{2 - 10}{-9 + 1} \), the focus shifts to calculating distances and direction using simple arithmetic rather than visual estimations.

Linear Equations: Interpreting the Slope

Linear equations form the backbone of algebra, often represented as \( y = mx + b \), where \(m\) is the slope, and \(b\) is the y-intercept. The slope, specifically, describes how the \(y\)-value, or dependent variable, changes with respect to the \(x\)-value, which is independent.
To understand how this fits into linear equations, notice from the solution that the slope \( m = 1 \), indicates a consistent, linear relationship: for every unit increase in \(x\), \(y\) increases by the same unit.
This constant rate of change is what makes this relationship linear, and graphically you would see a straight line that ascends at an angle of 45 degrees above the horizon. This knowledge is invaluable for analyzing trends, patterns, and predictions in fields ranging from economics to physics.