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Chapter 12: Problem 3

Give the equation and graph for a line with \(y\) -intercept equal to 3 and slope equal to -1 .

Short Answer

Expert verified
Answer: The equation of the line is y = -x + 3. The graph will have a y-intercept of 3 and a slope of -1, which means for every 1 unit move to the right, the line goes down by 1 unit.

Step by step solution

01

Identify the given information

We are given the \(y\)-intercept which is 3 and the slope which is -1. This means we have the point (0, 3) and the slope, m = -1.
02

Apply the point-slope form

We will use the point-slope form of a linear equation: \(y - y_1 = m(x - x_1)\). Substitute the point (0, 3) and the slope m = -1 to the equation: \(y - 3 = -1(x - 0)\).
03

Simplify the equation

Simplify the equation from step 2: \(y - 3 = -x\). Now add 3 to both sides of the equation: \(y = -x + 3\).
04

Write the final equation and create the graph

The final equation for the line is \(y = -x + 3\). To create the graph, plot the \(y\)-intercept point (0, 3). Then, use the slope m = -1 (which means going down 1 unit and moving to the right 1 unit) to find additional points on the line. Connect these points to create the line that represents the given equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The point-slope form of a linear equation is a useful tool to write an equation when you know a point on the line and the slope of the line. It is expressed as:
  • \(y - y_1 = m(x - x_1)\)
In this formula, \((x_1, y_1)\) represents a known point on the line and \(m\) represents the slope.
This form is particularly handy when dealing with circumstances where both slope and a single point are provided but the full equation is not immediately apparent.
Once you substitute the known values into the formula, you can either rearrange the equation to find the standard form, or directly use it for calculations. This is the starting point for solving the original exercise using the point-slope form to arrive at the known equation.
Slope-Intercept Form
The slope-intercept form is one of the most common ways to write the equation of a straight line. It takes the form:
  • \(y = mx + b\)
Here, \(m\) is the slope of the line, and \(b\) is the y-intercept (the point where the line crosses the y-axis). With this form, it's easy to understand both the direction of the line and where it starts on the graph.
For the original exercise, using the information given -- that the slope \(m\) is -1, and the y-intercept \(b\) is 3 -- we simply substitute these into the slope-intercept form to get:
\(y = -x + 3\).
This form allows for quick and accurate graphing as it reveals the intercept and slope at a glance, making it intuitive to visualize the line's trajectory.
Graphing Linear Equations
Once you have the linear equation in slope-intercept form, graphing it involves a few simple steps. Let's assume you have the equation:
  • \(y = -x + 3\)

1. Start by plotting the y-intercept on the graph. In this case, it's the point (0, 3).
2. Use the slope to find additional points. Here, the slope \(-1\) means for every one unit you move horizontally to the right, you move one unit down.
3. From the y-intercept point (0, 3), move one unit to the right and one unit down to locate another point on the line, like (1, 2).
4. Draw a straight line through these points extending across both ends of the graph.
By following these steps, you create a visual representation that accurately reflects the equation, making it easier to understand the relationship between variables in the equation.

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Most popular questions from this chapter

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