Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the fundamental concepts of the rectangular coordinate system and how it is used to plot and identify points. Students learn to determine locations by utilizing ordered pairs, to verify if points satisfy linear equations, and to create tables of solutions. The skills developed here form the basis for graphing linear equations and understanding the relationship between algebraic expressions and their graphical representations.

Learning Objectives

1

Plot points accurately on a rectangular coordinate system and identify their corresponding coordinates.

2

Determine the location of points within quadrants and on the axes including the origin.

3

Verify ordered pairs as solutions to a given linear equation in two variables.

4

Complete tables of solutions for linear equations by substituting values for x or y.

5

Graph linear equations by recognizing the relationship between algebraic equations and their plotted points on the coordinate plane.

Key Concepts

CONCEPT

DEFINITION

Rectangular Coordinate System

A system that uses two perpendicular number lines (the x-axis and y-axis) to determine the positions of points in a plane.

Ordered Pair

A pair of numbers (x, y) that represents the coordinates of a point, where x is the horizontal value and y is the vertical value.

Quadrant

One of the four regions created by the intersection of the x-axis and y-axis; numbered I (top right), II (top left), III (bottom left), and IV (bottom right).

Origin

The point where the x-axis and y-axis intersect, represented by the ordered pair (0, 0).

Linear Equation in Two Variables

An equation that can be written in the form Ax + By = C, representing a straight line with infinitely many solutions (ordered pairs).

Slope-Intercept Form

A form of a linear equation written as y = mx + b, where m represents the slope and b the y-intercept.

Example Problems

Example 1

Plot each point on a coordinate grid. $(3,2)$

Example 2

Plot each point on a coordinate grid. $(4,1)$

Example 3

Plot each point on a coordinate grid. $(1,5)$

Example 4

Plot each point on a coordinate grid. $(3,4)$

Example 5

Plot each point on a coordinate grid. $(4,1),(1,4)$
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Step-by-Step Explanations

QUESTION

How do you plot the point (2, 5) on a rectangular coordinate system?

STEP-BY-STEP ANSWER:

Step 1: Locate the x-coordinate (2) on the horizontal x-axis.
Step 2: Draw a light vertical line through x = 2.
Step 3: Locate the y-coordinate (5) on the vertical y-axis.
Step 4: Draw a light horizontal line through y = 5.
Step 5: The intersection of these two lines is the point (2, 5).
Final Answer: The point (2, 5) is plotted at the intersection of the vertical line through x = 2 and the horizontal line through y = 5.

Plotting a Point on the Coordinate System

QUESTION

Is the ordered pair (0, 3) a solution to the equation 3x + 2y = 6?

STEP-BY-STEP ANSWER:

Step 1: Substitute x = 0 and y = 3 into the equation: 3(0) + 2(3).
Step 2: Simplify the expression: 0 + 6 = 6.
Step 3: Compare the result to the right-hand side of the equation: 6 = 6.
Final Answer: Yes, (0, 3) is a solution to the equation since the left-hand side equals 6.

Verifying a Solution for a Linear Equation

QUESTION

How do you complete a table of solutions for y = 5x - 1?

STEP-BY-STEP ANSWER:

Step 1: Choose a value for x (commonly x = 0, then x = 1, etc.).
Step 2: Substitute the chosen x-value into the equation to solve for y.
Step 3: For x = 0, y = 5(0) - 1 = -1. Write the ordered pair (0, -1).
Step 4: For x = 1, y = 5(1) - 1 = 4. Write the ordered pair (1, 4).
Step 5: For x = 2, y = 5(2) - 1 = 9. Write the ordered pair (2, 9).
Final Answer: The table includes the points (0, -1), (1, 4), and (2, 9), which are solutions to the equation y = 5x - 1.

Completing a Table for a Linear Equation

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Common Mistakes

  • Reversing the order of coordinates when plotting ordered pairs (treating (x, y) as (y, x)).
  • Misidentifying quadrants by not properly considering the signs of the coordinates.
  • Failing to connect points correctly or not recognizing the need for arrows on the line to indicate it continues infinitely.
  • Incorrectly substituting values into equations leading to errors in verifying solutions.