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Chapter 6: Problem 82

Use a table of values to graph the equation. $$3 x+4 y=20$$

Short Answer

Expert verified
The graph of the equation \(3 x + 4 y = 20\) is a straight line that passes through the points (-2, 6.5), (0, 5), (2, 3.5), (4, 2), and (6, 0.5), which are obtained by substituting different x-values into the equation and solving for y.

Step by step solution

01

Choosing Values for x

Choose different values for \(x\). For this exercise, take five values (-2, 0, 2, 4, and 6)
02

Solve for Corresponding y Values

Substitute the chosen \(x\) values into the equation and solve for \(y\). The equation will look something like this: \(4y = 20 - 3x\) which simplifies to \(y = 5 - \frac{3}{4}x\).
03

Fill the Table

Create a table and fill it with the chosen \(x\) values and their corresponding \(y\) values. The table will look something like this: \[\begin{array}{|c|c|} \hline x & y \\ \hline -2 & 6.5 \\ 0 & 5 \\ 2 & 3.5 \\ 4 & 2 \\ 6 & 0.5 \\ \hline \end{array}\]
04

Plot the Points

Use the pairs (x, y) in the table to plot points on the graph: (-2, 6.5), (0, 5), (2, 3.5), (4, 2), and (6, 0.5). Then, draw a line that passes through all the points. This line graphically represents the equation.
05

Validate Your Graph

Choose another point that is not on your table, substituting those values for \(x\) and \(y\). If the numbers satisfy the equation, then your graph should be correct.

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Key Concepts

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

Table of Values
Understanding how to graph linear equations begins with creating a table of values. This table is essentially a list of paired numbers, each consisting of an input (often represented as 'x') and an output (usually 'y').

Steps for creating a table of values:
  • Select convenient values for 'x'. In our example, we chose values like -2, 0, 2, 4, and 6.
  • Substitute each selected 'x' value in the equation to find its corresponding 'y'.
This list of (x, y) pairs provides the necessary points to be plotted on a graph. It's essential to choose x-values that make the calculations simple and reveal the trend of the line clearly.
Plotting Points
Once you have a table of values, the next step is plotting each pair as a point on a graph. Each point is represented by its coordinates, which are the 'x' and 'y' values you calculated.

How to plot:
  • Find the 'x' value on the horizontal axis of your graph.
  • From this 'x' position, move vertically to the corresponding 'y' value on the graph's vertical axis.
  • Mark this position with a point.
In our example, points like (-2, 6.5) and (4, 2) are plotted. For a linear equation, these points should all line up in a straight line, demonstrating the linear nature of the equation.
Solving for y
In order to plot points on a graph efficiently, it's often easier to solve the linear equation for 'y'. This rearranges the equation to make 'y' a function of 'x'.

Steps to solve for 'y':
  • Start with the original equation: 3x + 4y = 20.
  • Isolate 'y' by subtracting '3x' from both sides, resulting in: 4y = 20 - 3x.
  • Divide every term by 4, giving: y = 5 - \(\frac{3}{4}\)x.
Now, with 'y' expressed in terms of 'x', it simplifies the process of finding 'y' values for your table. This also highlights the slope and intercept of the line, which can be useful for graphing and understanding the equation's behavior.
Linear Equations
Linear equations like 3x + 4y = 20 represent a straight line on a graph. They have a constant rate of change, meaning the relationship between 'x' and 'y' remains consistent.

Key characteristics of linear equations:
  • The standard form is often Ax + By = C, where A, B, and C are constants.
  • The line's slope can be found from the equation. In the rearranged form y = mx + b (where 'm' is the slope), it dictates how steep the line is.
  • The y-intercept is the point where the line crosses the y-axis. This can also be found in the equation: it's the value of 'b' in the form y = mx + b.
Understanding these properties helps in graphing and analyzing linear relationships, making predictions, or validating results from plotted graphs.

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