Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. (-3,10) and (5,-6)

Step-by-Step Solution

Answer

The line's equation is \(y = -2x + 4\).

Step 1: Identify the points and represent them as coordinates

The two points provided are (-3, 10) and (5, -6). We can label them as \((x_1, y_1) = (-3, 10)\) and \((x_2, y_2) = (5, -6)\). This information will help us find the slope of the line.

Step 2: Calculate the slope (m) of the line

The formula to calculate the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, we have:\[m = \frac{-6 - 10}{5 + 3} = \frac{-16}{8} = -2\].Thus, the slope of the line is \(-2\).

Step 3: Use the point-slope formula

The point-slope formula is \(y - y_1 = m(x - x_1)\). Let's use point (-3, 10) and the slope obtained:\[y - 10 = -2(x + 3)\].

Step 4: Simplify to slope-intercept form (y=mx+b)

To convert the equation from point-slope form to slope-intercept form, distribute and simplify:\[y - 10 = -2x - 6\].Adding 10 to both sides gives:\[y = -2x + 4\].This is the equation in slope-intercept form.

Key Concepts

Point-Slope Formula

The point-slope formula is a powerful tool in coordinate geometry. It is used to determine the equation of a line when you know a point on the line and its slope. The formula is given by:\( y - y_1 = m(x - x_1) \)Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) refers to the slope. This formula emphasizes how each point

• contributes to the overall structure of the line
• shows how the slope dictates its direction and steepness

The concept hinges on the idea that a line's infinite points are uniformly distributed along its path, with the slope maintaining this even distribution. Understanding the point-slope formula makes it easier to derive other forms of a linear equation, especially when switching between various coordinate representations.

Slope-Intercept Form

Once you've established the equation through the point-slope formula, converting it to slope-intercept form makes it simpler to visualize and interpret. The slope-intercept form is:\( y = mx + b \)In this form, \(m\) represents the line's slope, while \(b\) is the y-intercept — the point where the line crosses the y-axis. Here are some key insights:

• It allows for quick assessment of how steep the line is and which direction it leans.
• The y-intercept gives an entry point for plotting the line on a graph.

When you have a linear equation in this format, predicting values or comparing line equations becomes more straightforward. It also clarifies how a change in \(x\) results in a proportional change in \(y\), following the line's slope.

Coordinate Geometry

Coordinate geometry bridges algebra and geometry through the lens of a Cartesian plane. It provides a framework to analyze and understand lines and curves using coordinates. The core idea is locating points with ordered pairs — \((x, y)\). In the context of linear equations:

• The slope \(m\) tells us about the line's inclination or decline.
• Points like \((x_1, y_1)\) provide anchor positions to determine a line's possible paths.

With coordinate geometry, one can graph equations, determine distances between points, and even find the intersection points of lines. The discipline revolves around using algebraic expressions to solve geometric problems and vice versa, integrating visual learning with analytical thinking. Understanding these principles enhances the ability to interpret real-world data graphically.