Problem 81 Use a table of values to graph t... [FREE SOLUTION]

Chapter 6: Problem 81

Use a table of values to graph the equation. $$-2 x+2 y=5$$

Short Answer

Expert verified

The graph of the equation $-2x + 2y = 5$ is a straight line that passes through the points $(-1, 1.5), (0, 2.5), (1, 3.5), (2, 4.5)$

Step by step solution

Rewrite The Equation in y=mx+b Form

First, transform the equation $ -2x + 2y = 5 $ into slope-intercept form, $y = mx + b$, where m is the slope and b is the y-intercept. The given equation can be rewritten as $y = x + 2.5$

Constructing The Table

Choose a few reasonable values for x and substitute them in the equation $y = x + 2.5$ to obtain the corresponding values for y. For instance, we can take x = -1, 0, 1 and 2.

Compute Corresponding Y Values

Substitute the chosen x values in the equation to get y values. For instance, if x = -1, $y = -1 + 2.5 = 1.5$. If x = 0, $y = 0 + 2.5 = 2.5$. If x = 1, $y = 1 + 2.5 = 3.5$. If x = 2, $y = 2 + 2.5 = 4.5$.

Plotting The Graph

Plot the pairs of x and y values on the plane as points. The pairs are $(-1, 1.5), (0, 2.5), (1, 3.5), (2, 4.5)$. After plotting these points, draw a line that passes through them.

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Key Concepts

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

Slope-Intercept Form

When dealing with linear equations, an important form to understand is the slope-intercept form. This is written as $ y = mx + b $, where $ m $ represents the slope of the line, and $ b $ is the y-intercept鈥攖he point where the line crosses the y-axis. Understanding this form gives you a quick insight into the line's behavior:

The slope $ m $ shows the steepness and direction of the line. A positive slope means the line rises as $ x $ increases, while a negative slope means it falls.
The y-intercept $ b $ tells you where on the y-axis the line starts. This is the value of $ y $ when $ x = 0 $.

For our example, the equation $ -2x + 2y = 5 $ was transformed to $ y = x + 2.5 $, which is now in slope-intercept form. Here, $ m = 1 $ and $ b = 2.5 $. This indicates the line has a positive slope, going up one unit in $ y $ for each unit it moves across in $ x $, and it crosses the y-axis at 2.5.

Table of Values

Using a table of values is a helpful way to graph equations. This approach allows you to choose different values for $ x $, calculate the corresponding $ y $ values, and get points you can plot. To create a table of values, follow these easy steps:

Select several values for $ x $. It's often useful to choose simple numbers such as -1, 0, 1, 2, etc.
Substitute each $ x $ value into your equation to solve for $ y $.

For instance, with $ y = x + 2.5 $, choosing $ x = -1, 0, 1, $ and $ 2 $, we get:

$ x = -1 $ gives $ y = 1.5 $
$ x = 0 $ gives $ y = 2.5 $
$ x = 1 $ gives $ y = 3.5 $
$ x = 2 $ gives $ y = 4.5 $

Thus, your table of values will provide you with coordinate pairs $(-1, 1.5), (0, 2.5), (1, 3.5), (2, 4.5)$ that are crucial for plotting your graph.

Plotting Points

Plotting points on a graph is a simple yet essential step in visualizing the behavior of an equation. Each coordinate pair from your table of values gives you a specific point on the graph where $ x $ is the horizontal position, and $ y $ is the vertical position.Here鈥檚 how to plot points:

Start with the first coordinate pair $(-1, 1.5)$. Find $ -1 $ on the x-axis, and $ 1.5 $ on the y-axis, and mark that point.
Repeat this for each pair: $(0, 2.5), (1, 3.5), (2, 4.5)$.
Make sure the points are accurately placed based on your scale for the axes.

Once all points are plotted, the next step is to draw a line through them. The line should extend across the graph, ensuring that it maintains the consistent pattern determined by the slope of the equation. This way, you've successfully graphed a linear equation using individual points derived from your table of values.

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91影视

Use a table of values to graph the equation. $$-2 x+2 y=5$$

Short Answer

Step by step solution

Rewrite The Equation in y=mx+b Form

Constructing The Table

Compute Corresponding Y Values

Plotting The Graph

Key Concepts

Slope-Intercept Form

Table of Values

Plotting Points

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