Problem 78 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ (-1,4), \quad(-2,-4) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is y = 8x + 12.

Step 1: Find the slope (m) of the line using the given points

Using point-slope formula, we can find the slope of the line: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using the given points \((-1, 4)\) and \((-2, -4)\): $$m = \frac{-4 - 4}{-2 - (-1)} = \frac{-8}{-1} = 8$$

Step 2: Write the general equation of the line using one of the points and the slope

We can use the point-slope form equation to get the general equation of the line: $$\textit{Point-slope form:} \ y - y_1 = m(x - x_1)$$ Using point \((-1, 4)\) and the slope \(m = 8\), we get: $$ y - 4 = 8(x - (-1)) $$

Step 3: Convert the equation to the slope-intercept form

Now we simplify and rewrite the equation in the slope-intercept form \(y = mx + b\): \begin{aligned} y - 4 &= 8(x + 1) \\ y - 4 &= 8x + 8 \\ y &= 8x + 8 + 4 \\ y &= 8x + 12 \end{aligned} In slope-intercept form, the equation of the line is \(y = 8x + 12\).

Key Concepts

Slope Calculation

The slope of a line is an essential concept in algebra that helps us understand how much a line rises or falls as it moves from left to right along the coordinate plane. When calculating the slope (\(m\)) of a line that passes through two points, we use the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points.
For example, given the points \((-1, 4)\) and \((-2, -4)\), we can substitute these values into the slope formula.

• \(y_2 = -4\), \(y_1 = 4\), \(x_2 = -2\), \(x_1 = -1\)
• \(m = \frac{-4 - 4}{-2 - (-1)} = \frac{-8}{-1} = 8\)

This means for every unit of run, the line rises 8 units. Positive slope indicates an upward trend from left to right.

Point-Slope Form

Point-slope form is invaluable when you know one point on a line and its slope. The point-slope form equation is:

• \(y - y_1 = m(x - x_1)\)

Here, \(m\) represents the slope, and \((x_1, y_1)\) is a known point on the line.
Utilizing the point \((-1, 4)\) and the calculated slope \(m = 8\), the point-slope form becomes:

• \(y - 4 = 8(x - (-1))\)
• This expands to \(y - 4 = 8(x + 1)\)

The point-slope form is flexible because it captures the line's behavior using any point from the line.
Useful for quickly writing an equation when the line is described by its slope and any point.

Equation of a Line

Understanding the equation of a line in different forms helps with various algebraic tasks. The slope-intercept form is particularly popular because it clearly shows the slope and the y-intercept:

• \(y = mx + b\)

Where \(m\) is the slope and \(b\) is the y-intercept, or where the line crosses the y-axis.
Converting from point-slope form to slope-intercept form involves simplifying the equation:

• Start with \(y - 4 = 8(x + 1)\)
• Expand to \(y - 4 = 8x + 8\)
• Then, add 4 to both sides to isolate \(y\): \(y = 8x + 8 + 4\)

The final equation in slope-intercept form is \(y = 8x + 12\), revealing that the line rises 8 units for every unit it moves to the right and crosses the y-axis at 12.