Problem 122 - Elementary and Intermediate Algebra

Question

Consider the line that passes through \(P(-2,3)\) and \(Q(4,-4)\). Write the equation in slope-intercept form of line \(P Q\).

Step-by-Step Solution

Answer

The equation of line PQ in slope-intercept form is \\(y = -\frac{7}{6}x + \frac{2}{3}\\).

Step 1: Calculate the Slope

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For the points \(P(-2, 3)\) and \(Q(4, -4)\), the slope \(m\) is \(-\frac{4 + 3}{4 + 2} = -\frac{7}{6}\). So, the slope of the line is \(-\frac{7}{6}\).

Step 2: Use the Point-Slope Formula

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Using point \(P(-2, 3)\) and the slope \(m = -\frac{7}{6}\), substitute into the formula: \(y - 3 = -\frac{7}{6}(x + 2)\).

Step 3: Expand and Simplify the Equation

Expand the equation \(y - 3 = -\frac{7}{6}(x + 2)\) to \(y - 3 = -\frac{7}{6}x - \frac{7}{3}\). Add 3 to both sides to isolate \(y\): \(y = -\frac{7}{6}x + \frac{2}{3}\).

Step 4: Write in Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. From the previous step, we have \(y = -\frac{7}{6}x + \frac{2}{3}\), which is in slope-intercept form.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness. Imagine a hill—the steeper the hill, the greater the slope. To find the slope of a line that runs between two points, you need to find the difference in the y-values and the x-values of those points. This difference is like finding rise over run.
The formula for calculating the slope (\(m\)) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Here, \(x_1, y_1\) are the coordinates of the first point, and \(x_2, y_2\) are the coordinates of the second point. Just subtract the y-coordinates and x-coordinates of your points and divide them to get the slope.

• If the line slopes upwards from left to right, \(m\) is positive.
• If the line slopes downwards from left to right, \(m\) is negative.
• If the line is perfectly horizontal, meaning no rise, \(m = 0\).
• For a vertical line, the slope is undefined.

Point-Slope Formula

The point-slope formula is another way to write a linear equation, especially useful when you know a point on the line and the slope. This formula is:
\[y - y_1 = m(x - x_1)\]
Here's what each part means:

• \(y\) and \(x\) are variables representing any point on the line.
• \(x_1, y_1\) are the coordinates of a known point on the line.
• \(m\) is the slope of the line.

Using this formula, you can plug in the slope and the point you already know, and you'll get another form of the line equation.
Then, you can transform it into slope-intercept form by further simplifying, which is helpful to see the slope and the y-intercept directly.

Linear Equation

A linear equation represents a straight line when graphed on a coordinate plane. It's one of the simplest forms of math equations and appears frequently in real-world applications. The most common form you'll come across is the slope-intercept form:
\[y = mx + b\]
Here's a breakdown of this form:

• \(y\) and \(x\) are the variables that represent any point on the line.
• \(m\) is the slope of the line, showing how steep the line is.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

By putting the equation into this form, you can quickly identify both the slope and y-intercept without having to calculate them separately. Linear equations in this form make it easy to sketch the graph of the line or to understand how one variable changes with respect to another.