Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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724,371 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section teaches the method of substitution as an effective strategy for solving systems of equations. Key steps include isolating a variable in one equation, substituting into the other, solving for the remaining variable, and verifying the solution through back-substitution. The approach is applicable to both linear and nonlinear systems and is vital for real-world problems like investment distribution, break-even analysis, and ticket sales comparisons. Graphical interpretations further strengthen the understanding by demonstrating the relationship between algebraic solutions and intersection points on graphs.

Learning Objectives

1

Apply the method of substitution to solve systems of equations, including both linear and nonlinear systems.

2

Demonstrate how to isolate a variable, substitute into the other equation, and use back-substitution to obtain the complete solution.

3

Interpret the graphical meaning of solutions to systems of equations and relate them to real-life modeling, such as break?even analysis and investment problems.

4

Check solutions for systems by substituting the obtained values into the original equations to ensure accuracy.

Key Concepts

CONCEPT

DEFINITION

System of Equations

A set of two or more equations with common variables that are solved simultaneously.

Ordered Pair

A pair of numbers representing the values of variables (e.g., (x, y)) that satisfies all equations in the system.

Method of Substitution

A procedure for solving systems where one equation is solved for one variable and then substituted into another equation to reduce the system to a single variable.

Back-Substitution

The process of substituting the value found for one variable back into an earlier equation to find the value of the other variable.

Nonlinear System

A system in which at least one equation is nonlinear, such as including quadratic or logarithmic terms.

Break-Even Point

The point at which total cost equals total revenue, often determined as the intersection of two functions in applied problems.

Example Problems

Example 1

Fill in the blanks. A set of two or more equations in two or more variables is called a ________ of ________.

Example 2

Fill in the blanks. A ________ of a system of equations is an ordered pair that satisfies each equation in the system.

Example 3

Fill in the blanks. Finding the set of all solutions to a system of equations is called ________ the system of equations.

Example 4

Fill in the blanks. The first step in solving a system of equations by the method of ________ is to solve one of the equations for one variable in terms of the other variable.

Example 5

Fill in the blanks. Graphically, the solution of a system of two equations is the ________ of ________ of the graphs of the two equations.
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Step-by-Step Explanations

QUESTION

Solve the system: Equation 1: y = 2x – 3; Equation 2: 3x + y = 7.

STEP-BY-STEP ANSWER:

Step 1: Notice that Equation 1 already expresses y in terms of x: y = 2x – 3.
Step 2: Substitute the expression for y into Equation 2: 3x + (2x – 3) = 7.
Step 3: Combine like terms: 3x + 2x = 5x, so the equation becomes 5x – 3 = 7.
Step 4: Add 3 to both sides: 5x = 10.
Step 5: Divide both sides by 5 to solve for x: x = 2.
Step 6: Substitute x = 2 back into Equation 1 to find y: y = 2(2) – 3 = 4 – 3 = 1.
Final Answer: The solution is the ordered pair (2, 1).

Solving a Linear System by Substitution

QUESTION

Solve the system: Equation 1: y = x^2 – x – 1; Equation 2: x – y = 2.

STEP-BY-STEP ANSWER:

Step 1: From Equation 2, express y in terms of x: y = x – 2.
Step 2: Substitute y = x – 2 into Equation 1: x – 2 = x^2 – x – 1.
Step 3: Rearrange the equation to one side: 0 = x^2 – x – 1 – (x – 2) = x^2 – 2x + 1.
Step 4: Factor the quadratic: (x – 1)^2 = 0.
Step 5: Solve for x: x = 1.
Step 6: Substitute x = 1 back into y = x – 2 to find y: y = 1 – 2 = -1.
Final Answer: The solution is the ordered pair (1, -1).

Solving a Nonlinear System by Substitution

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Common Mistakes

  • Failing to correctly isolate the variable before substitution, leading to algebraic errors.
  • Arithmetic errors during combining like terms or applying the distributive property.
  • Not back-substituting to verify the solution, which may allow errors to go unnoticed.
  • Confusing the substitution method with elimination or graphical methods, particularly in nonlinear systems.
  • Overlooking the possibility of no real solution when the substitution leads to a contradiction or an unsolvable equation.