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Chapter 3: Problem 33

Graph each equation by using the slope and y-intercept. $$ 2 x+y=-5 $$

Short Answer

Expert verified
y = -2x - 5. Plot points (0, -5) and (1, -7) and draw the line.

Step by step solution

01

- Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of an equation is given by \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Rearrange the given equation \( 2x + y = -5 \) to this form. Subtract \( 2x \) from both sides to get \[ y = -2x - 5 \].
02

- Identify the Slope and y-Intercept

From the equation \( y = -2x - 5 \), identify the slope \( m \) and the y-intercept \( b \). The slope \( m \) is -2 and the y-intercept \( b \) is -5.
03

- Plot the y-Intercept

Start by plotting the y-intercept \( b \) on the graph. Since \( b = -5 \), place a point on the y-axis at (0, -5).
04

- Use the Slope to Plot Another Point

The slope \( m = -2 \) can be interpreted as \( -2/1 \), meaning rise of -2 and run of 1. From the point (0, -5), move down 2 units and to the right 1 unit to plot the next point at (1, -7).
05

- Draw the Line

Draw a straight line through the points (0, -5) and (1, -7). This is the graph of the equation \( y = -2x - 5 \).

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Key Concepts

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

Slope-Intercept Form
To graph a linear equation easily, we use the slope-intercept form of a line equation, which is given by: \[ y = mx + b \] It consists of two main components: the slope (\( m \)) and the y-intercept (\( b \)). The form helps us understand how to graph the equation. First, we convert the given equation to this form. For example, with the equation \( 2x + y = -5 \), we rearrange it to: \( y = -2x - 5 \).
Slope
The slope (\( m \)) tells us the steepness of the line and the direction it goes. It is the ratio of the change in y (vertical change) to the change in x (horizontal change). In the equation \( y = -2x - 5 \), the slope is -2. This indicates that for every unit increase in x, y decreases by 2 units. We can interpret this as a fraction: \( -2/1 \), which means we go down 2 units vertically and 1 unit to the right horizontally.
Y-Intercept
The y-intercept (\( b \)) is where the line crosses the y-axis. For the equation \( y = -2x - 5 \), the y-intercept is -5. It tells us that when x is 0, y is -5. To plot this point on the graph, place a dot at the coordinates (0, -5). This is the starting point for graphing a line using the slope-intercept form.
Plotting Points
Once we have the y-intercept, we can plot another point using the slope. Starting from (0, -5), use the slope \( -2/1 \). Move down 2 units and 1 unit to the right to get the next point (1, -7). Plot this point on the graph. Repeat the process as needed. The more points you plot, the more accurate your graph will be.
Linear Equations
A linear equation is any equation that graphs to a straight line. It can be written in several forms, but the slope-intercept form \( y = mx + b \) is especially useful for graphing. Linear equations are fundamental in algebra and appear in many real-world applications. They are known for their constant rate of change, represented visually by a straight line on a graph. Understanding how to work with them will help you tackle more complex math problems.

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Most popular questions from this chapter

Graph each equation by using the slope and y-intercept. $$ y=3 x+2 $$

Graph each equation by using the slope and y-intercept. $$ y=-\frac{1}{2} x+2 $$

Do \((3,4)\) and \((4,3)\) correspond to the same point in the plane? Explain.

A point \((x, y)\) has the property that \(x y>0 .\) In which quadrant(s) must the point lie? Explain.

What is the slope of a line whose graph is parallel to the graph of \(3 x+y=7 ?\) Perpendicular to the graph of \(3 x+y=7 ?\)
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