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Chapter 1: Problem 55

Make a table of values, and sketch the graph of the equation. $$4 x+5 y=40$$

Short Answer

Expert verified
Table points: (0, 8), (5, 4), (10, 0). Connect them to form a line.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

First, let's rewrite the given equation, \(4x + 5y = 40\), in slope-intercept form \(y = mx + b\). Solve for \(y\): 1. Subtract \(4x\) from both sides: \[5y = -4x + 40\]2. Divide every term by 5: \[y = -\frac{4}{5}x + 8\].Now we have the equation in the form \(y = -\frac{4}{5}x + 8\).
02

Select Convenient Values for x

To make a table of values, choose convenient values for \(x\). Let's select integers that are easy to work with, such as \(x = 0\), \(x = 5\), and \(x = 10\).
03

Calculate Corresponding y Values

Using the equation \(y = -\frac{4}{5}x + 8\), calculate the \(y\) values for each chosen \(x\):- For \(x = 0\): \[y = -\frac{4}{5}(0) + 8 = 8\]- For \(x = 5\): \[y = -\frac{4}{5}(5) + 8 = -4 + 8 = 4\]- For \(x = 10\): \[y = -\frac{4}{5}(10) + 8 = -8 + 8 = 0\].Now we have the points \((0, 8)\), \((5, 4)\), and \((10, 0)\).
04

Create a Table of Values

Organize these points into a table:\[\begin{array}{c|c}x & y \\hline0 & 8 \5 & 4 \10 & 0 \\end{array}\]
05

Sketch the Graph

Plot the points \((0, 8)\), \((5, 4)\), and \((10, 0)\) on a coordinate plane. Then, draw a straight line through these points to represent the graph of the equation. This line should slope downwards from left to right due to the negative slope \(-\frac{4}{5}\).

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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 an essential concept in algebra for expressing straight-line equations. This form is given by the formula \(y = mx + b\). Here:
  • \(m\) represents the slope of the line. The slope indicates the steepness and direction of the line. For example, a slope of \(-\frac{4}{5}\) suggests the line descends as it moves from left to right.
  • \(b\) is the y-intercept, where the line crosses the y-axis. In our example, \(b = 8\) means the line intersects the y-axis at \(y = 8\).
Being comfortable with converting equations like \(4x + 5y = 40\) into the slope-intercept form helps in understanding and graphing them easily. Once rewritten as \(y = -\frac{4}{5}x + 8\), it becomes straightforward to identify the slope and y-intercept, which are crucial for graphing.
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of a horizontal x-axis and a vertical y-axis that intersect at the origin \((0, 0)\).
  • The x-axis runs left to right, and the y-axis runs up and down.
  • Points on this plane are determined by ordered pairs \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.
Understanding a coordinate plane is crucial because it allows for the visualization of mathematical concepts, including the intersection of equations and geometric interpretations. Graphing equations like \(y = -\frac{4}{5}x + 8\) becomes easier when using a coordinate plane to see how the line behaves across different sections.
Table of Values
Creating a table of values is a practical method to understand how a linear equation behaves for different x-values. Here鈥檚 how you can create one:
  • Select specific x-values. In our exercise, we chose \(x = 0\), \(x = 5\), and \(x = 10\) for convenience.
  • Substitute these x-values into the equation to calculate corresponding y-values.
For example, with the equation \(y = -\frac{4}{5}x + 8\):
  • If \(x = 0\), then \(y = 8\)
  • If \(x = 5\), then \(y = 4\)
  • If \(x = 10\), then \(y = 0\)
These calculated pairs \((0, 8)\), \((5, 4)\), and \((10, 0)\) help create a table, which serves as a guide for plotting the line on the graph.
Graphing Linear Equations
Graphing linear equations involves plotting points and connecting them to form a straight line. It helps in visualizing the relationship between variables. To graph the equation \(y = -\frac{4}{5}x + 8\):
  • Begin by plotting the points you obtained in your table of values on the coordinate plane: \((0, 8)\), \((5, 4)\), and \((10, 0)\).
  • Use these points to draw a straight line through them. This line represents the equation \(y = -\frac{4}{5}x + 8\).
  • The line should slope downward from left to right, consistent with the slope \(-\frac{4}{5}\).
By following these steps, you visually express the solution set of the equation, providing a clear image of how x and y values relate.

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Most popular questions from this chapter

Gas Mileage The gas mileage \(q\) (measured in milgal) for a particular vehicle, driven at \(v\) mi/h, is given by the formula \(g=10+0.9 v-0.01 v^{2},\) as long as \(v\) is between \(10 \mathrm{mi} / \mathrm{h}\) and \(75 \mathrm{mi} / \mathrm{h}\). For what range of speeds is the vehicle's mileage 30 milgal or better?

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