Problem 40 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (-2,4),(3,-5) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is $$y = -\frac{9}{5}x + \frac{2}{5}.$$

Step 1: Find the slope

We will use the formula $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ to find the slope of the line passing through the given points \((-2,4)\) and \((3,-5)\). Plugging in the coordinates, we get $$ m = \frac{-5 - 4}{3 - (-2)} = \frac{-9}{5}. $$

Step 2: Find the y-intercept (b)

To find the y-intercept, we will use the slope-intercept form equation and one of the given points (either point can be used). We will use point \((-2, 4)\). First, plug the values into the equation: $$ y = mx + b, $$ $$ 4 = \left(-\frac{9}{5}\right)(-2) + b. $$

Step 3: Solve for b

Now we can solve for b: $$ 4 = \frac{18}{5} + b. $$ Subtract \(\frac{18}{5}\) from both sides to isolate 'b': $$ b = 4 - \frac{18}{5} = \frac{2}{5}. $$

Step 4: Write the equation

Now that we have both the slope and the y-intercept values, we can write the equation of the line in slope-intercept form: $$ y = mx + b $$ $$ y = -\frac{9}{5}x + \frac{2}{5}. $$ So, the equation of the line passing through points \((-2,4)\) and \((3, -5)\) is $$ y = -\frac{9}{5}x + \frac{2}{5}. $$

Key Concepts

Understanding Linear Equations

Linear equations are essential in mathematics and describe lines in a plane. They tell you how one variable depends on another.
This relationship is shown in a straight-line graph. The general formula for a linear equation is given by:

• \(y = mx + b\)

In this format:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

In problems where you need to find the equation of a line, this formula helps to plug in specific points to determine the equation of that line accurately.
Understanding how to use linear equations in the slope-intercept form is crucial for analyzing relationships in algebra and graphing.

Finding the Slope of a Line

The slope of a line is a measure of how steep the line is. To find the slope, you need two points that lie on the line.
The formula for calculating slope \(m\) is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
This formula tells us the change in \(y\) divided by the change in \(x\), often referred to as "rise over run." In the example from the exercise, the two points used are \((-2,4)\) and \((3,-5)\).
Plugging in these points we get:

• \(m = \frac{-5 - 4}{3 - (-2)} = \frac{-9}{5}\).

This negative slope indicates that the line slants downwards as it moves from left to right. Remembering this simple formula makes slope calculations straightforward and manageable!

Calculating the Y-intercept

The y-intercept \(b\) is the value at which the line crosses the y-axis. To find \(b\), first use the slope-intercept form of a linear equation. With the slope \(m\) found, select one point from the provided data to solve for \(b\).
Let's see how it works with a point \((-2, 4)\) and slope \(m = \frac{-9}{5}\):

• Start with the formula \(y = mx + b\).
• Insert slope \(m\) and coordinates \((-2, 4)\): \(4 = \left(-\frac{9}{5}\right)(-2) + b\).
• Calculate for \(b\): \(4 = \frac{18}{5} + b\).
• Convert and solve: \(b = 4 - \frac{18}{5} = \frac{2}{5}\).

So, the y-intercept is \(\frac{2}{5}\). This point tells us exactly where the line intersects the y-axis.
With the slope \( m = \frac{-9}{5} \) and y-intercept \( b = \frac{2}{5} \), the linear equation can be fully expressed! Understanding y-intercept calculations is key to completing linear equations in slope-intercept form.