Problem 22 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((0,-8)\) and \((4,0)\)

Step-by-Step Solution

Answer

The slope-intercept form is \\(y = 2x - 8\\).

Step 1: Identify the Given Points

We are given the points \(0, -8\) and \(4, 0\). These points are used to find the equation of the line.

Step 2: Find the Slope of the Line

The formula for the slope (\(m\)) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \((m = \frac{y_2 - y_1}{x_2 - x_1})\). Substitute the given points: \(m = \frac{0 + 8}{4 - 0} = \frac{8}{4} = 2\). Thus, the slope of the line is 2.

Step 3: Use the Point-Slope Formula

The point-slope form of a line is \((y - y_1 = m(x - x_1))\). We can use the point \(0, -8\) (you could also use \(4, 0\), but choosing \(0, -8\) makes it straightforward as it is the y-intercept). Substitute the known values: \(y - (-8) = 2(x - 0)\). Simplify: \(y + 8 = 2x\).

Step 4: Convert to Slope-Intercept Form

The slope-intercept form is \(y = mx + b\). From \(y + 8 = 2x\), solve for \(y\): \(y = 2x - 8\). Hence, the slope-intercept form is \(y = 2x - 8\).

Key Concepts

Slope Calculation

Calculating the slope of a line is a fundamental concept in coordinate geometry. Let's break it down simply. The slope, often denoted as \(m\), is a measure of how steep a line is. It tells us how much the line rises for a certain horizontal distance. Think of it like climbing hills. Steeper hills mean bigger slopes.

To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use this formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

It's crucial to remember the order of the points. Subtraction is sensitive to order!

In the given problem, we have points \((0, -8)\) and \((4, 0)\). Substituting these into our formula, we have:

• \(m = \frac{0 + 8}{4 - 0} = \frac{8}{4} = 2\)

So the slope of the line is **2**, which means for every unit you move to the right, the line goes up by 2 units.

Equation of a Line

Once we have the slope, we often want to write the equation of a line. The equation of a line can tell us every single point that the line passes through. There are different forms, but here we'll focus on the slope-intercept form which is very popular.

The slope-intercept form is expressed as:

• \(y = mx + b\)

Here, \(m\) is the slope, and \(b\) is the y-intercept (the point where the line crosses the y-axis).

To find \(b\), let's start with the point-slope form, which is:

• \(y - y_1 = m(x - x_1)\)

Using the point \((0, -8)\) and replacing \(m\) with 2, we have:

• \(y - (-8) = 2(x - 0)\)

Simplifying, \(y + 8 = 2x\). When we solve for \(y\), we get \(y = 2x - 8\). This is our line in slope-intercept form, with a slope of 2 and a y-intercept of -8.

Coordinate Geometry

Coordinate geometry is the magic that lets us use algebra to explore geometric concepts on a flat surface called the coordinate plane. It blends together lines, points, and slopes beautifully.

Imagine a grid with intersecting lines. Each intersection is a point, defined by an \((x, y)\) pair, like a GPS coordinate you might use to find a location. The x-axis is the horizontal line and the y-axis is the vertical one. This whole grid is a playground for lines and shapes.

In our exercise, we plotted two points: \((0, -8)\) and \((4, 0)\). Drawing a line through these points on the coordinate plane gives us a visual of our linear equation, \(y = 2x - 8\).

• The slope (2) tells us that as x increases by 1 unit, y increases by 2 units.
• The line crosses the y-axis at -8, our y-intercept.

By understanding these coordinates, we're able to communicate the nature and direction of our line clearly. It's like blending a map with algebra to see how everything ties together in geometry.