Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

Group icon
40,131 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Section 8.1 focuses on solving systems of linear equations using the substitution and elimination methods. It highlights how to form systems from real-world problems such as ticket sales and financial planning, and discusses the nature of consistent, inconsistent, and dependent systems. Understanding these concepts not only aids in solving algebraic problems but also helps in interpreting graphs of lines or planes in geometric contexts.

Learning Objectives

1

Solve systems of linear equations using the substitution method.

2

Solve systems of linear equations using the elimination method.

3

Identify and categorize systems as consistent (independent), inconsistent, or dependent.

4

Apply algebraic techniques to solve systems involving two or three variables.

5

Interpret graphical representations of systems and relate them to their algebraic solutions.

Key Concepts

CONCEPT

DEFINITION

System of Linear Equations

A collection of two or more linear equations that share the same set of variables.

Linear Equation

An equation that can be written in the form a1x1 + a2x2 + … + anxn = b, where the variables are raised only to the first power.

Substitution Method

A technique for solving systems of equations in which one solves one of the equations for one variable and substitutes this expression into the other equation(s).

Elimination Method

A method for solving systems of equations by adding or subtracting the equations after appropriately multiplying them in order to eliminate one of the variables.

Consistent System

A system of equations that has at least one solution.

Inconsistent System

A system of equations that has no solution; graphically, the lines are parallel.

Dependent System

A system of equations with infinitely many solutions; the equations represent the same line (or plane) graphically.

Example Problems

Example 1

Solve the equation: $3 x+4=8-x$

Example 2

(a) Graph the line: $3 x+4 y=12$ (b) What is the slope of a line parallel to this line?

Example 3

If a system of equations has no solution, it is said to be _________.

Example 4

If a system of equations has one solution, the system is ________ and the equations are _______.

Example 5

If the solution to a system of two linear equations containing two unknowns is $x=3, y=-2,$ then the lines intersect at the point ________.
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Solve the system using substitution: 2x + y = 5 and -4x + 6y = 12.

STEP-BY-STEP ANSWER:

Step 1: Solve equation (1) for y: y = 5 - 2x.
Step 2: Substitute y = 5 - 2x into equation (2): -4x + 6(5 - 2x) = 12.
Step 3: Distribute and simplify: -4x + 30 - 12x = 12.
Step 4: Combine like terms: -16x + 30 = 12.
Step 5: Subtract 30 from both sides: -16x = -18.
Step 6: Divide both sides by -16: x = 9/8.
Step 7: Substitute x = 9/8 back into y = 5 - 2x: y = 5 - 2*(9/8) = 5 - 9/4 = 11/4.
Final Answer: x = 9/8, y = 11/4.

Substitution Method

QUESTION

Solve the system using elimination: 2x + 3y = 1 and -x + y = -3.

STEP-BY-STEP ANSWER:

Step 1: Multiply equation (2) by 2 to align coefficients of x: -2x + 2y = -6.
Step 2: Add the new equation (2) to equation (1): (2x + 3y) + (-2x + 2y) = 1 + (-6).
Step 3: Combine like terms: 5y = -5.
Step 4: Solve for y: y = -1.
Step 5: Substitute y = -1 back into one of the original equations (use equation (2)): -x + (-1) = -3, so -x = -2 and x = 2.
Final Answer: x = 2, y = -1.

Elimination Method

Scroll left
Scroll right

Common Mistakes

  • Making sign errors when isolating variables or substituting expressions.
  • Failing to correctly multiply equations when aligning coefficients in the elimination method.
  • Not checking the final solution in all original equations, leading to overlooked errors.
  • Confusing inconsistent systems with dependent systems, and vice versa.