Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

32,192 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This chapter provides a foundational understanding of graphing in the rectangular coordinate system. Key takeaways include learning how to accurately plot points and interpret their positions by identifying quadrants, verifying solutions to linear equations through substitution, and generating tables of solutions to effectively graph linear equations. Mastering these skills is crucial for further studies in algebra and real-world data analysis.

Learning Objectives

Plot points accurately in the rectangular coordinate system and identify their corresponding quadrants.

Verify solutions to linear equations in two variables by substituting ordered pairs.

Complete tables of solutions for linear equations by choosing appropriate values for x (or y) and solving for the other variable.

Graph linear equations by plotting points and connecting them to form a line, while understanding the relationship between an equation's solutions and its graph.

Key Concepts

CONCEPT

DEFINITION

Rectangular Coordinate System

A system using two perpendicular number lines (the x-axis and y-axis) to define points in the plane by ordered pairs.

Ordered Pair

A pair of numbers (x, y) that represent the coordinates of a point in the rectangular coordinate system. The order is important: the first number is the x-coordinate and the second is the y-coordinate.

Origin

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

Quadrants

The four regions created by the intersection of the x-axis and y-axis, labeled I, II, III, and IV, each defined by the positive or negative signs of the coordinates.

Linear Equation in Two Variables

An equation of the form Ax + By = C, where A and B are not both zero. Its graph is a straight line with infinitely many solutions.

Standard Form

A format of a linear equation written as Ax + By = C, typically with integer coefficients and A ≥ 0.

Example Problems

Example 1

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. (a) (-4,2) (b) (-1,-2) (c) (3,-5) (d) (-3,5) (e) $\left(\frac{5}{3}, 2\right)$

Example 2

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. (@) (-2,-3) (b) (3,-3) (c) (-4,1) (d) (4,-1) (e) $\left(\frac{3}{2}, 1\right)$

Example 3

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. (a) (3,-1) (b) (-3,1) (c) (-2,2) (d) (-4,-3) (e) $\left(1, \frac{14}{5}\right)$

Example 4

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. (a) (-1,1) (b) (-2,-1) (c) (2,1) (d) (1,-4) (e) $\left(3, \frac{7}{2}\right)$

Example 5

In the following exercises, plot each point in a rectangular coordinate system. (@) (-2,0) (b) (-3,0) (c) (0,0) (d) (0,4) (e) (0,2)

Step-by-Step Explanations

QUESTION

How do you plot the point (1, 3) on a rectangular coordinate system?

STEP-BY-STEP ANSWER:

Step 1: Find the value 1 on the x-axis (horizontal axis).
Step 2: Find the value 3 on the y-axis (vertical axis).
Step 3: From x = 1, draw a light vertical line and from y = 3, draw a light horizontal line.
Step 4: Mark the intersection point of these two lines.
Final Answer: The point (1, 3) is plotted at the intersection of the vertical line through 1 and the horizontal line through 3.

Plotting a Point

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Substitute x = 2 and y = 0 into the equation: 3(2) + 2(0).
Step 2: Compute the left-hand side: 6 + 0 = 6.
Step 3: Compare the computed value with the right-hand side of the equation, which is 6.
Final Answer: Since both sides are equal (6 = 6), (2, 0) is a solution to the equation 3x + 2y = 6.

Verifying a Linear Equation Solution

Common Mistakes

Mixing up the order of coordinates when writing an ordered pair (e.g., writing (y, x) instead of (x, y)).
Confusing the roles of the x-axis and y-axis, leading to incorrect point placement.
Overlooking the significance of sign conventions in determining the quadrant of a point.
Arithmetic errors when substituting values into equations, which can lead to incorrect verification of solutions.