Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on solving systems of equations using various methods: substitution, graphing, and elimination. Students learn to isolate variables, substitute expressions, and use graphing utilities to identify intersections. These techniques not only provide multiple approaches to obtaining solutions, but they also extend to modeling real-life problems, from break-even analysis in business to complex investment scenarios. The key takeaway is that understanding the relationships between equations and the geometry of their graphs is crucial in both academic and real-world applications.

Learning Objectives

1

Understand how to determine the slope of a line and use it to write linear equations.

2

Apply the substitution method to solve systems of equations in two variables.

3

Utilize graphing techniques to identify the point(s) of intersection representing the solution of a system.

4

Employ the elimination method to conveniently solve systems by canceling a variable.

5

Model real-life problems, such as break-even analysis and investment scenarios, using systems of equations.

Key Concepts

CONCEPT

DEFINITION

System of Equations

A set of two or more equations with the same set of variables whose common solution is the set of values satisfying every equation simultaneously.

Solution

An ordered pair (or tuple) that makes every equation in the system true when its components are substituted for the variables.

Substitution Method

A technique for solving systems by isolating one variable in one equation and substituting the resulting expression into the other equation.

Graphing Method

A method that involves plotting the equations on the coordinate plane; the point(s) where the graphs intersect represent the solution(s) of the system.

Elimination Method

A method where corresponding terms of the equations are added or subtracted to eliminate one variable, simplifying the system to a single equation in one variable.

Slope-Intercept Form

A way of writing a linear equation in the form y = mx + b, where m is the slope and b is the y-intercept.

Example Problems

Example 1

Fill in the blank(s). A set of two or more equations in two or more unknowns is called a______of _____.

Example 2

Fill in the blank(s). A_____of a system of equations is an ordered pair that satisfies each equation in the system.

Example 3

Fill in the blank(s). The first step in solving a system of two equations in $x$ and $y$ by the method of ____ is to solve one of the equations for one variable in terms of the other.

Example 4

Fill in the blank(s). A point of intersection of the graphs of the equations of a system is a _____ of the system.

Example 5

What is the point of intersection of the graphs of the cost and revenue functions called?
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Step-by-Step Explanations

QUESTION

Solve the system: { x + y = 4, x - y = 2 }.

STEP-BY-STEP ANSWER:

Step 1: Solve the first equation for y. y = 4 - x.
Step 2: Substitute y = 4 - x into the second equation: x - (4 - x) = 2.
Step 3: Simplify the equation: x - 4 + x = 2, which gives 2x - 4 = 2.
Step 4: Add 4 to both sides: 2x = 6, then divide by 2 to obtain x = 3.
Step 5: Substitute x = 3 back into y = 4 - x to find y: y = 4 - 3 = 1.
Final Answer: The solution is (3, 1).

Substitution Method Example

QUESTION

Solve the system by graphing: { y = 4 - x, y = x - 2 }.

STEP-BY-STEP ANSWER:

Step 1: Write both equations in slope-intercept form. They are already in that form.
Step 2: Plot the line y = 4 - x using its slope (-1) and y-intercept (4).
Step 3: Plot the line y = x - 2 using its slope (1) and y-intercept (-2).
Step 4: Identify the point of intersection of the two lines on the graph, which is (3, 1).
Step 5: Verify by substituting (3, 1) into both equations.
Final Answer: The solution is (3, 1).

Graphing Method Example

QUESTION

Solve the system: { 3x + 2y = 4, 5x - 2y = 8 }.

STEP-BY-STEP ANSWER:

Step 1: Notice that the y-terms have coefficients that are opposites (2y and -2y).
Step 2: Add the equations: (3x + 2y) + (5x - 2y) = 4 + 8, which simplifies to 8x = 12.
Step 3: Divide by 8 to find x: x = 12 / 8 = 3/2.
Step 4: Substitute x = 3/2 into one of the original equations (e.g., 3x + 2y = 4): 3(3/2) + 2y = 4 gives 9/2 + 2y = 4.
Step 5: Solve for y: 2y = 4 - 9/2 = (8/2 - 9/2) = -1/2, so y = -1/4.
Final Answer: The solution is (3/2, -1/4).

Elimination Method Example

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Common Mistakes

  • Failing to properly isolate a variable before substitution, leading to algebraic errors.
  • Not checking the obtained solution in the original equations.
  • Mistakenly combining terms during elimination, especially with negative signs.
  • Relying solely on the graphical method without verifying solutions algebraically.
  • Overlooking the fact that systems can have one solution, no solution, or infinitely many solutions.