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Magazine Chapter 7: Problem 28
Graph each equation by plotting ordered pairs. $$y=x+5$$
Short Answer
Expert verified
The graph includes points (-2, 3), (0, 5), and (2, 7) forming a straight line.
Step by step solution
01
Understanding the Equation
This is a linear equation in the slope-intercept form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For \( y = x + 5 \), the slope \( m = 1 \) and the y-intercept \( b = 5 \).
02
Choosing Values for x
Select three different values for \( x \) to find ordered pairs. Let's choose \( x = -2, 0, 2 \).
03
Calculating y for Each x
Substitute the chosen \( x \) values into the equation \( y = x + 5 \) to find corresponding \( y \) values. For \( x = -2 \), \( y = -2 + 5 = 3 \); for \( x = 0 \), \( y = 0 + 5 = 5 \); and for \( x = 2 \), \( y = 2 + 5 = 7 \).
04
Writing the Ordered Pairs
The ordered pairs derived from the calculations are \((-2, 3)\), \((0, 5)\), and \((2, 7)\).
05
Plotting the Ordered Pairs
On a graph, plot the points \((-2, 3)\), \((0, 5)\), and \((2, 7)\).
06
Drawing the Line
Connect the plotted points with a straight line. The line represents the graph of the equation \( y = x + 5 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a linear equation is a simple yet powerful tool. It takes the format \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. These elements provide clear information about the line's behavior and where it crosses the y-axis.
- Slope \( m \): This defines the line's steepness. A positive slope means the line rises from left to right, while a negative slope means it falls. When the slope is 1, as in our equation \( y = x + 5 \), it means the line rises one unit up for every unit it moves to the right.
- Y-intercept \( b \): This is the point where the line crosses the y-axis. For \( y = x + 5 \), the line crosses at \( (0, 5) \), showing you where the graph starts in relation to the y-axis.
Ordered Pairs
In mathematics, ordered pairs are an essential concept for plotting graphs. An ordered pair is a pair of numbers written in a specific order, usually as \((x, y)\). These numbers tell us the exact location of a point on a Cartesian coordinate system.
To find ordered pairs for the equation \( y = x + 5 \):
To find ordered pairs for the equation \( y = x + 5 \):
- Choose values for \( x \). This is known as an independent variable since its value can be freely chosen.
- Calculate the corresponding \( y \) values by substituting \( x \) into the equation. In our example, \( y \) is calculated as \( y = x + 5 \).
- Form the ordered pairs. For instance, if \( x = -2 \), then \( y = 3 \), giving us the ordered pair \((-2, 3)\).
Plotting Points
Plotting points is a crucial skill for visualizing linear equations on a graph. It involves marking specific locations, identified by ordered pairs, on the coordinate grid.
To plot the points for \( y = x + 5 \):
To plot the points for \( y = x + 5 \):
- Start by locating the x-coordinate of each ordered pair on the horizontal axis.
- Then, find the corresponding y-coordinate on the vertical axis.
- Place a dot where the two coordinates meet. For the ordered pair \((-2, 3)\), you would place a dot where \( x = -2 \) and \( y = 3 \).
