Problem 9 - Algebra 2

Question

What is an equation of the line through \((2,-4)\) and \((-3,-1) ?\) A. \(y=-\frac{3}{5} x+\frac{26}{5}\) B. \(y=-\frac{3}{5} x-\frac{14}{5}\) C. \(y=\frac{3}{5} x-\frac{26}{5}\) D. \(y=\frac{3}{5} x+\frac{14}{5}\)

Step-by-Step Solution

Answer

The equation is Option B: \(y = -\frac{3}{5}x - \frac{14}{5}\).

Step 1: Calculate the Slope

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[m = \frac{y_2 - y_1}{x_2 - x_1}\.\]Using the points \((2, -4)\) and \((-3, -1)\), we have:\[m = \frac{-1 - (-4)}{-3 - 2} = \frac{-1 + 4}{-5} = \frac{3}{-5} = -\frac{3}{5}\.\]

Step 2: Use the Point-Slope Form Equation

Now that we have the slope, use the point-slope form of a line equation: \[ y - y_1 = m(x - x_1) \]Choose one of the given points, say \((2, -4)\), and substitute the slope \(m = -\frac{3}{5}\):\[ y + 4 = -\frac{3}{5}(x - 2) \].

Step 3: Simplify the Equation

Distribute the slope on the right-hand side:\[ y + 4 = -\frac{3}{5}x + \frac{6}{5} \].To solve for \(y\), subtract 4 from both sides:\[ y = -\frac{3}{5}x + \frac{6}{5} - 4 \].Convert 4 to a fraction with a denominator of 5:\[ y = -\frac{3}{5}x + \frac{6}{5} - \frac{20}{5} \].Combine fractions:\[ y = -\frac{3}{5}x - \frac{14}{5} \].

Step 4: Identify the Correct Answer

The simplified equation of the line is:\[ y = -\frac{3}{5}x - \frac{14}{5} \], which corresponds to Option B.

Key Concepts

Slope Calculation

In coordinate geometry, understanding how to calculate the slope of a line is crucial. The slope is a measure of how steep a line is. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. The formula for the slope, given two points

• \((x_1, y_1)\)
• \((x_2, y_2)\)

on a line, is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, you subtract the corresponding coordinates to find the difference in the y-values and the x-values. Then, you divide these differences to find the slope. This tells us the direction and the steepness of the line. A positive slope means the line rises as you move from left to right. Conversely, a negative slope means it falls. In our exercise, between the points (2, -4) and (-3, -1), the slope was calculated as \(-\frac{3}{5}\), indicating the line descends moderately.

Point-Slope Form

Once you have the slope of a line, you can easily write the line's equation using the point-slope form. This form is especially helpful when you know the slope and one point on the line. The point-slope form of an equation is represented as:\[ y - y_1 = m(x - x_1) \]where:

• \(m\) is the slope of the line,
• \((x_1, y_1)\) is a point on the line.

Simply substitute the known slope and coordinates of the point into the equation. For example, using point (2, -4) and the slope \(-\frac{3}{5}\), the equation becomes:\[y + 4 = -\frac{3}{5}(x - 2)\].Point-slope form is a robust method to determine the line’s equation efficiently when you have a point and the slope.

Simplifying Equations

Simplifying equations is an essential skill in math, especially when dealing with linear equations. The goal is to rearrange the equation into a simpler or more standard form, often the slope-intercept form \[ y = mx + b \]After substituting the known values into the point-slope form, distribute the slope through the expression and then rearrange to solve for \(y\). For example, given:\[y + 4 = -\frac{3}{5}(x - 2)\]distributing \(-\frac{3}{5}\) yields:\[y + 4 = -\frac{3}{5}x + \frac{6}{5}\].Next, subtract 4 from both sides to isolate \(y\):\[y = -\frac{3}{5}x + \frac{6}{5} - 4\].Convert 4 into a fraction that allows for subtraction with \(\frac{6}{5}\), \(\frac{20}{5}\), and find:\[y = -\frac{3}{5}x - \frac{14}{5}\].Through these steps, the equation becomes easier to analyze, making it clearer to identify solutions or further explore graphing.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a crucial branch of mathematics. It connects algebra and geometry using a coordinate system. By plotting points, lines, and shapes in a plane, you can utilize algebraic methods to solve geometric problems. Understanding the equation of a line is fundamental in this area. In coordinate geometry, the equation of a line not only defines it but also provides information such as slope, direction, and intercepts. By using coordinates, this form of geometry allows for precise calculations and can handle complex shapes easily. In our exercise, we used two points to find the equation of the line, demonstrating how coordinate geometry helps us understand and solve real-world mathematical problems. Visualizing how points and lines correspond can make math problems seem more tangible and manageable. With these principles, you can derive properties of shapes and their relationships on a coordinate plane.