Problem 14 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { The slope is }-2 \text {, and the } x \text { intercept is } 4 \text {. } $$

Step-by-Step Solution

Answer

The equation is y = -2x + 8 .

Step 1: Understand the problem

The problem provides a slope of -2 and an x-intercept of 4. The goal is to find the equation of the line with these characteristics.

Step 2: Recall the slope-intercept form

The slope-intercept form of a linear equation is given by y = mx + b , where m is the slope and b is the y-intercept.

Step 3: Set up the equation with the given slope

Since the slope ( m ) is given as -2, the equation becomes y = -2x + b .

Step 4: Use the x-intercept to find the y-intercept

An x-intercept of 4 means the point (4, 0) lies on the line. Substitute x = 4 and y = 0 into the equation y = -2x + b to find b .

Step 5: Solve for the y-intercept (b)

Substitute (4, 0) into y = -2x + b : 0 = -2(4) + b Simplify: 0 = -8 + b Add 8 to both sides of the equation: b = 8.

Step 6: Write the final equation

Now that b is found, substitute b back into the equation: y = -2x + 8 .

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is an essential concept in algebra. It is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form is incredibly useful because it allows us to quickly identify the slope and y-intercept just by looking at the equation. Knowing the slope, we can understand how steep the line is and in which direction it trends. The y-intercept tells us where the line crosses the y-axis, which is key to graphing the line accurately.

finding y-intercept

To find the y-intercept, you need at least one point on the line and the slope. In the given problem, the x-intercept is provided as 4. This point is written as (4, 0).
First, substitute this point and the slope into the slope-intercept form. Our equation currently looks like this \(y = -2x + b\). We'll use the coordinates of the given point (4, 0):
Substitute 4 for \(x\), and 0 for \(y\):
\(0 = -2(4) + b\)
Simplify to find \(b\): \(0 = -8 + b\)
Adding 8 to both sides, we get \(b = 8\). Now, we know that the y-intercept is 8.

solving linear equations

Solving linear equations involves finding the values of the variables that make the equation true. Here, we solve for the y-intercept using substitution. With an equation in the form \(y = mx + b\), if we know the slope \(m\) and have a point \((x_1, y_1)\), we substitute these values into the equation to solve for \(b\).
Let's recap how we did this: after identifying the slope \(m = -2\) and the x-intercept \(4\), we used these in our equation:
\(0 = -2(4) + b\)
This equation simplifies to \(0 = -8 + b\), and we solve for \(b\) to find \(b = 8\). Thus, the equation of the line becomes \(y = -2x + 8\).

graphing linear equations

Graphing linear equations involves plotting points and drawing a line through them. Start with the y-intercept, which we found to be 8. This means the line crosses the y-axis at (0, 8). Next, use the slope to find another point.
The slope of -2 indicates that for every 1 unit you move to the right along the x-axis, you'll move 2 units down along the y-axis. Starting from (0, 8), move right by 1 unit to (1, 6), since going down 2 units from 8 gives you 6.
Plot these points: (0, 8) and (1, 6). Draw a line through them. This represents the equation \(y = -2x + 8\).
Always double-check by verifying that other points on the line satisfy the equation.