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Chapter 0: Problem 7

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$x+y=5$$

Short Answer

Expert verified
Plot points (0,5) and (1,4), then draw a line through them.

Step by step solution

01

Write the Equation in Slope-Intercept Form

The standard form of the equation is given as: \[x + y = 5\]To convert it to the slope-intercept form (\(y = mx + b\)), solve for \(y\): \[y = 5 - x\]
02

Identify the Slope and Y-Intercept

From the equation \(y = 5 - x\), identify the slope (\(m\)) and the y-intercept (\(b\)).\(m = -1\) and \(b = 5\).
03

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is where the line crosses the y-axis. Here, \(b = 5\), so plot a point at (0,5).
04

Use the Slope to Find Another Point

The slope of the line is -1, which means for every 1 unit you move to the right on the x-axis, move 1 unit down on the y-axis.From the point (0,5), move 1 unit to the right (to x=1) and 1 unit down (to y=4), plotting the point at (1,4).
05

Draw the Line

With the two points (0,5) and (1,4) plotted, draw a straight line through them. This line represents the equation \(x + y = 5\). Ensure your line extends to the edge of your graph in both directions.

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Key Concepts

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

Slope-Intercept Form
When graphing linear equations, the slope-intercept form is very helpful because it provides an easy way to identify the slope and y-intercept. The general format for slope-intercept form is:

y = mx + b

Here, m represents the slope of the line, and b represents the y-intercept, which is where the line crosses the y-axis. This form is particularly useful because you can rewrite almost any linear equation into this format. For example, given the equation x + y = 5, solving for y will give you:

y = 5 - x.

Now, it's in slope-intercept form where m = -1 and b = 5.
Slope
Slope is a measure of how steep a line is. In the slope-intercept form equation y = mx + b, the slope is represented by m. If m is positive, the line will rise as it moves from left to right, and if m is negative, the line will fall.

For example, in the equation y = 5 - x, the slope m is -1. This means that for each unit you move to the right on the x-axis, you move 1 unit down on the y-axis. This negative slope indicates that the line is decreasing.

To visualize this, start from a particular point and follow the slope to find other points on the line.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. In the equation y = mx + b, the y-intercept is given by b.

For example, in y = 5 - x, the y-intercept b is 5. This means that the line crosses the y-axis at the point (0, 5).

To graph this, simply start by plotting a point where the y-axis and 5 intercepts. From here, you can use the slope to find other points on the line. Remember, the y-intercept is essential in giving you a starting point for your graph.
Plotting Points
Plotting points is the process of marking points on a graph to visualize a line. Start by finding the y-intercept, then use the slope to find additional points.

For example, with the equation y = 5 - x, start by plotting the y-intercept (0, 5). Next, use the slope of -1 to find other points. Starting from (0, 5), move 1 unit to the right and 1 unit down to reach the point (1, 4).

By repeating this process, you create a series of points that form a straight line. Finally, draw a line through these points, ensuring it extends across the entire graph.

With these steps, you can accurately graph any linear equation by hand.

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