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Chapter 1: Problem 17

Graph each equation in Exercises \(13-28\). Let \(x=-3,-2,-1,0\) \(1,2,\) and 3 $$ y=2 x+1 $$

Short Answer

Expert verified
To graph the equation y=2x+1, we create a table of values using the given x-values (-3,-2,-1,0,1,2,3), substitute them into the equation to get y-values, plot these points on a graph paper and then draw a line through them.

Step by step solution

01

Table of Values

The first step is to create a table of values using the given x-values and the equation. By substituting each x-value into the equation, we will get the corresponding y-value. For instance, if x=-3, then y=2(-3)+1, which gives y=-5. We repeat this process for each x-value to get the corresponding y-values.
02

Plot the Ordered Pairs

Now that the table of values is ready, we plot these ordered pairs (x, y) on a graph paper. It is important to remember the pairs represent the co-ordinates on the graph. For example, if x=-3 and y=-5, this gives us the point (-3,-5).
03

Draw the Line

Once all the values are plotted, we draw a line that best fits the points. Since this is a linear equation, the points should form a straight line. The plotted line represents the solution to the given linear equation.

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Key Concepts

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

Equation of a Line
An equation of a line is a mathematical statement that describes how two variables, usually "x" and "y," are related. In this context, the equation given is \( y = 2x + 1 \).
This is a linear equation because it forms a straight line when graphed.
The key components of the equation \( y = 2x + 1 \) are:
  • Slope: The coefficient of "x," which is 2, represents the slope of the line. This tells us the line rises two units for every unit it moves to the right.
  • Y-intercept: The constant term, 1, indicates where the line crosses the y-axis. This is the point (0,1) on the graph.
Understanding the equation helps in easily predicting and plotting the line's behavior on the graph. Each part of the equation plays a crucial role in shaping the graph.
Table of Values
A table of values is used to organize information from the equation systematically. It involves choosing certain x-values, plugging them into the equation, and determining the corresponding y-values.
For example, in the equation \( y = 2x + 1 \):
  • If \( x = -3 \), \( y = 2(-3) + 1 = -5 \).
  • If \( x = 0 \), \( y = 2(0) + 1 = 1 \).
  • If \( x = 3 \), \( y = 2(3) + 1 = 7 \).
By repeating this for each given x-value, we create a table of pairs, where each pair is an ordered pair \((x, y)\).
This table provides a clear guide for plotting points later, turning the abstract equation into concrete data.
Plotting Points
Plotting points is the act of marking points on a graph that correspond to the x and y values from the table of values.
Each ordered pair \((x, y)\) represents a point on the Cartesian coordinate system:
  • The x-coordinate tells you how far left or right to go, from the origin (0,0).
  • The y-coordinate tells you how far up or down to go, from the origin.
For example, the point (-3, -5) means starting at the origin, moving three units left, and five units down. After plotting all points, they should ideally lie on a straight line if the equation is linear. This visual representation on a graph provides clarity and confirms the relationship described by the equation.
Linear Equations in Algebra
Linear equations are among the most foundational concepts in algebra. These equations are called "linear" because their graph is a straight line.
They generally take the form \( y = mx + b \) or \( ax + by = c \).
Specific characteristics of linear equations include:
  • They do not include exponents greater than one on x or y.
  • They represent a constant rate of change, as shown by the slope.
  • They are used to model real-world situations where relationships between two variables are proportional.
By understanding linear equations, students can unlock the ability to analyze various patterns in data. They also form the basis for more advanced mathematical concepts, making them an essential building block in a student's education.

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Most popular questions from this chapter

In Exercises \(123-124,\) list all numbers that must be excluded from the domain of each rational expression. $$ \frac{7}{2 x^{2}-8 x+5} $$

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y=5 x^{2}+3 x \text { and } y=2 $$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=2 k w+ \quad 2 l h+ \quad 2 w h \text { for } h$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$\frac{1}{p}+\frac{1}{q}=\frac{1}{f} \text { for } f$$

The rectangular painting in the figure shown measures 12 inches by 16 inches and is surrounded by a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 72 inches. Determine the width of the frame.
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