Problem 10 - Calculus Single Variable

Question

Write the point-slope equation of the line determined by the two given points. (12,1),(-4,-4)

Step-by-Step Solution

Answer

\( y - 1 = \frac{5}{16}(x - 12) \)

Step 1: Calculate Slope

To find the slope (\( m \)), we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the given points (12,1) and (-4,-4): \( m = \frac{-4 - 1}{-4 - 12} = \frac{-5}{-16} = \frac{5}{16} \).

Step 2: Choose Point for Point-Slope Form

You can choose either of the two points to write the point-slope equation. Let's select the point (12,1).

Step 3: Write Point-Slope Equation

The point-slope form of a line is given by \( y - y_1 = m(x - x_1) \). Substituting \( m = \frac{5}{16} \), \( x_1 = 12 \), and \( y_1 = 1 \), we get: \( y - 1 = \frac{5}{16}(x - 12) \).

Key Concepts

Slope Calculation

To determine the equation of a line, one of the first steps is to calculate the slope. The slope, denoted as \( m \), is a measure of how steep a line is. It describes the rate of change between the two variables on the graph. To calculate the slope between two points, you use what is known as the "slope formula":\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. The formula subtracts the coordinates of the points—\(y\) values from each other, and \(x\) values from each other—to find how much the line "rises" or "falls" vertically per unit of horizontal movement.

• "Rise" is how much the vertical value (\(y\)) changes.
• "Run" is how much the horizontal value (\(x\)) changes.

In our example, using the points \((12,1)\) and \((-4,-4)\), we compute the slope as \( \frac{-4 - 1}{-4 - 12} = \frac{-5}{-16} = \frac{5}{16} \). The positive slope indicates the line rises from left to right.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves plotting points, lines, and shapes on the coordinate plane to solve geometric problems. This branch of mathematics allows us to represent geometric figures with algebraic equations, blending algebra and geometry harmoniously.The coordinate plane includes two axes:

• The horizontal axis, called the \(x\)-axis, represents all possible values for the independent variable \(x\).
• The vertical axis, called the \(y\)-axis, represents all possible values for the dependent variable \(y\).

Using these two axes, we can pinpoint any location in the plane with a coordinate pair \((x, y)\). This system enables us to easily understand relationships between points and lines by translating geometric problems into algebraic ones. In the exercise example, by plotting the points \((12,1)\) and \((-4,-4)\), we find a linear relationship through the computed slope.

Linear Equations

Linear equations describe a straight line on a coordinate plane and are foundational in understanding relationships between variables. These equations are typically written in one of several forms, the most common being the slope-intercept form \(y = mx + b\) and the point-slope form \(y - y_1 = m(x - x_1)\).In our exercise, we used the point-slope form of a line equation. The point-slope form is particularly useful when you know:

• A point on the line \((x_1, y_1)\).
• The slope \(m\).

Given the slope \(\frac{5}{16}\) and a point \( (12,1) \), we substitute these into the equation to get:\[ y - 1 = \frac{5}{16}(x - 12) \]This equation gives a clear and concise representation of the line connecting the two points. Linear equations make complex relationships between variables easier to understand, offering a simple model for predicting values along a line.