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Chapter 5: Problem 15

Complete the table of values and graph each equation. y=-2 x+3

Short Answer

Expert verified
We chose x-values -2, -1, 0, 1, and 2 and found the corresponding y-values using the equation \(y = -2x + 3\). The table of values is: x | y ---|--- -2 | 7 -1 | 5 0 | 3 1 | 1 2 | -1 We then plotted these points and drew the line representing the equation \(y = -2x + 3\).

Step by step solution

01

Chose x-values

Select several x-values to create a table of values. For example, choose x = -2, -1, 0, 1, and 2.
02

Calculate y-values

For each chosen x-value, calculate the corresponding y-value using the given equation (y = -2x + 3): When x = -2: y = -2(-2) + 3 = 4 + 3 = 7 When x = -1: y = -2(-1) + 3 = 2 + 3 = 5 When x = 0: y = -2(0) + 3 = 0 + 3 = 3 When x = 1: y = -2(1) + 3 = -2 + 3 = 1 When x = 2: y = -2(2) + 3 = -4 + 3 = -1
03

Create the table of values

Organize the x- and y-values calculated above into a table of values: x | y ---|--- -2 | 7 -1 | 5 0 | 3 1 | 1 2 | -1
04

Plot the points on the graph

Plot the points from the table of values on a graph: (-2, 7), (-1, 5), (0, 3), (1, 1), and (2, -1).
05

Draw the line

Connect the points plotted on the graph to create a line. This line is the graphical representation of the given equation, y = -2x + 3. Now the table of values is completed, and the graph of the equation y = -2x + 3 is drawn.

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Key Concepts

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

Table of Values
When dealing with linear equations, a table of values is a powerful tool. It helps to organize and visualize how different values of "x" affect the output, or "y". This method simplifies understanding the behavior of the equation. To create a table of values, you follow a straightforward process:
  • Select a few x-values. It can be beneficial to choose both negative and positive numbers, as well as zero. This gives a more complete view of the equation.
  • Substitute each chosen x-value into the equation to solve for y. This step might require some basic arithmetic, but it's a chance to see how the equation works in action.
  • Once you have corresponding y-values, list them in tandem with their x-values. The result is a neat table that shows the relationship between x and y for those selected points. For example, with the equation \(y = -2x + 3\), choosing values for x like -2, -1, 0, 1, and 2, results in corresponding y-values of 7, 5, 3, 1, and -1.
By using a table of values, equations become less abstract, providing concrete points that can later be used for graphing.
Graphing Linear Equations
Once the table of values is ready, the next step is to visualize the linear equation by graphing it. This begins by plotting each of the points from the table onto a graph. Each point corresponds to an (x, y) pair, representing a specific solution of the equation.
To effectively graph these linear equations, follow these steps:
  • Draw a set of axes on a graph paper or a digital graphing tool. The horizontal axis represents x-values, and the vertical axis represents y-values.
  • Take each pair of values from your table and plot them as a point on the graph. For example, the point \((-2, 7)\) is located 2 units to the left of the origin and 7 units above it.
  • Once all the points are plotted, connect them with a straight line. Linear equations should always form a straight line when graphed.
Graphing gives a visual perspective of how the linear equation behaves across different x-values. The line on the graph represents every possible solution to the equation. This visualization can reveal the rate at which y changes as x increases or decreases.
Slope-Intercept Form
The slope-intercept form is a format of linear equations that makes it easy to understand the line's characteristics. This form is written as \(y = mx + b\). Here:
  • \(m\) represents the slope of the line. It tells us the steepness and direction of the line. A positive slope means the line rises as x increases, whereas a negative slope means it falls.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis. This value denotes where x is zero.
For the equation \(y = -2x + 3\), \(-2\) is the slope. This means the line decreases by 2 units on the y-axis for every increase of 1 unit on the x-axis. The y-intercept is 3, indicating the line crosses the y-axis at \(y = 3\).
Understanding the slope-intercept form enables quick predictions about the graph's appearance. It's a key skill when interpreting or constructing graphs of linear equations.

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Most popular questions from this chapter

The perimeter of a triangle is \(29 \mathrm{cm}\). The longest side is \(5 \mathrm{cm}\) longer than the shortest side, and the sum of the two smaller sides is \(5 \mathrm{cm}\) more than the longest side. Find the lengths of the sides of the triangle.

(4.6) Determine if each pair of lines is parallel, perpendicular or neither. $$\begin{aligned} &x+2 y=22\\\ &2 x-y=0 \end{aligned}$$

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