Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section outlines two primary algebraic methods for solving systems of linear equations – substitution and elimination. Both methods are flexible and extend to systems with three variables. Recognizing the nature of the system – whether it is consistent with a unique solution, inconsistent (no solution), or dependent (infinitely many solutions) – is crucial. Practical applications, from ticket sales to curve fitting, highlight the significance of these techniques in modeling and real-world problem solving.

Learning Objectives

1

Solve systems of linear equations using the substitution method by isolating one variable and substituting into the other equation.

2

Apply the elimination method to remove one variable by aligning coefficients and adding or subtracting equations.

3

Identify and analyze different types of systems (consistent, inconsistent, and dependent) both algebraically and graphically.

4

Extend methods to solve systems involving three variables and understand their geometric interpretation as intersecting planes.

5

Utilize systems of equations in real-world applications such as ticket sales, curve fitting, and other modeling problems.

Key Concepts

CONCEPT

DEFINITION

System of Linear Equations

A collection of two or more linear equations involving the same set of variables whose solution is the set of variable values that satisfy every equation simultaneously.

Substitution Method

A strategy to solve a system of equations by solving one equation for one variable and substituting that expression into another equation.

Elimination Method

A method for solving systems by adding or subtracting equations after adjusting coefficients so that one variable cancels out.

Inconsistent System

A system of equations that has no solution because the equations represent parallel lines that never intersect.

Dependent System

A system with infinitely many solutions because the equations essentially represent the same line (or plane in higher dimensions).

Augmented Matrix

A matrix that represents a system of linear equations where the coefficients and constants are arranged in a compact form for use in algorithmic solution methods.

Example Problems

Example 1

The distance $d$ from $P_{1}=(2,-5)$ to $P_{2}=(4,-2)$ is d= ______

Example 2

To complete the square of $x^{2}-3 x,$ add _______.

Example 3

Find the intercepts of the equation $y^{2}=16-4 x^{2}$.

Example 4

The point that is symmetric with respect to the $y$ -axis to the point $(-2,5)$ is _____.

Example 5

To graph $y=(x+1)^{2}-4,$ shift the graph of $y=x^{2}$ to the (left/right) ______ unit(s) and then (up/down) unit (s).
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Step-by-Step Explanations

QUESTION

Solve the system: 2x + y = -1 and -4x + 6y = 42.

STEP-BY-STEP ANSWER:

Step 1: Solve the first equation for y. From 2x + y = -1, subtract 2x to get y = -2x - 1.
Step 2: Substitute y = -2x - 1 into the second equation: -4x + 6(-2x - 1) = 42.
Step 3: Distribute and simplify: -4x - 12x - 6 = 42, which simplifies to -16x - 6 = 42.
Step 4: Solve for x by adding 6 to both sides: -16x = 48, then dividing by -16 to obtain x = -3.
Step 5: Substitute x = -3 back into y = -2x - 1 to get y = -2(-3) - 1 = 6 - 1 = 5.
Final Answer: x = -3 and y = 5.

Solving by Substitution

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Multiply the second equation by 2 so that the coefficients of x are additive inverses: -2x + 2y = -6.
Step 2: Add the new equation to the first equation: (2x + 3y) + (-2x + 2y) = 1 + (-6), which yields 5y = -5.
Step 3: Solve for y: divide both sides by 5 to obtain y = -1.
Step 4: Substitute y = -1 into one of the original equations, for example, 2x + 3(-1) = 1, resulting in 2x - 3 = 1.
Step 5: Solve for x: add 3 to both sides getting 2x = 4, then divide by 2 to find x = 2.
Final Answer: x = 2 and y = -1.

Solving by Elimination

QUESTION

Solve the system: x + y - z = -1; 4x - 3y + 2z = 16; 2x - 2y - 3z = 5.

STEP-BY-STEP ANSWER:

Step 1: Use the first equation (x + y - z = -1) to eliminate x from equations (2) and (3). Multiply equation (1) by -4 and add to equation (2) to eliminate x.
Step 2: Multiply equation (1) by -2 and add to equation (3) to eliminate x from that equation.
Step 3: This yields a new system with two equations in y and z. Choose to eliminate z from these two equations by appropriate multiplication and addition.
Step 4: Solve for y from the resulting equation and back-substitute into one of the two-variable equations to find z.
Step 5: Substitute the found values of y and z into the original equation (1) to determine x.
Final Answer: x = 2, y = -2, and z = 1.

Solving a System of Three Variables

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Common Mistakes

  • Failing to properly isolate a variable during the substitution process, leading to algebraic errors.
  • Incorrectly aligning coefficients in the elimination method, which may result in the wrong cancellation of terms.
  • Forgetting to check the final solution in all original equations, missing inconsistent systems.
  • Misinterpreting dependent systems by not expressing the solution in parameterized form.
  • Overcomplicating a system without choosing the simplest equation to substitute or eliminate a variable.