Book cover for Elementary Algebra

Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

Group icon
32,192 Students Helped

Homework Questions

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

Summary

This section covers the fundamentals of solving systems of linear equations by graphing and substitution. It reinforces the idea that a system may have one solution, no solution, or infinitely many solutions depending on how the lines interact. Students learn to transform the equations into slope-intercept form, graph them accurately, and determine the point of intersection. The concepts of consistent versus inconsistent systems and independent versus dependent equations are also emphasized, providing a strong foundation for tackling real-world problems using systems of equations.

Learning Objectives

1

Determine whether a given ordered pair is a solution to a system of linear equations.

2

Solve systems of linear equations by graphing and classify the solution as unique, none, or infinitely many.

3

Identify and use various methods, such as substitution and graphical analysis, to solve systems of linear equations.

4

Translate real-world problems into systems of equations and solve them using appropriate methods.

Key Concepts

CONCEPT

DEFINITION

System of Linear Equations

A set of two or more linear equations that share the same variables and are considered simultaneously.

Solution to a System

An ordered pair (or tuple) that satisfies all equations in the system.

Graphing Method

A procedure that involves drawing the graphs of each linear equation on the same coordinate system and identifying their point(s) of intersection.

Consistent System

A system of equations that has at least one solution (unique solution or infinitely many).

Inconsistent System

A system of equations that has no solution, often represented by parallel lines in a graph.

Independent Equations

Equations that have distinct graphs (lines) intersecting at a single point, yielding one unique solution.

Dependent Equations

Equations that represent the same line, so every solution to one is a solution to the other, resulting in infinitely many solutions.

Slope-Intercept Form

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

Example Problems

Example 1

In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\{\begin{array}{l} 2 x-6 y=0 \\ 3 x-4 y=5 \end{array}\right. $$ (a) (3,1) (b) (-3,4)

Example 2

In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\{\begin{array}{l} 7 x-4 y=-1 \\ -3 x-2 y=1 \end{array}\right. $$ (a) (b) (1,-2)

Example 3

In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\{\begin{array}{l} 2 x+y=5 \\ x+y=1 \end{array}\right. $$ (a) (4,-3) (b) (2,0)

Example 4

In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\{\begin{array}{l} -3 x+y=8 \\ -x+2 y=-9 \end{array}\right. $$ (a) (-5,-7) (b) (-5,7)

Example 5

In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\{\begin{array}{l} x+y=2 \\ y=\frac{3}{4} x \end{array}\right. $$ (a) $\left(\frac{8}{7}, \frac{6}{7}\right)$ (b) $\left(1, \frac{3}{4}\right)$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Solve the system: y = 2x + 1 and y = 4x - 1 by graphing.

STEP-BY-STEP ANSWER:

Step 1: Identify the equations in slope-intercept form: the first equation is y = 2x + 1 (slope 2, intercept 1) and the second is y = 4x - 1 (slope 4, intercept -1).
Step 2: Graph both lines on the same coordinate plane using the slope and y-intercept for each line.
Step 3: Observe where the two lines intersect. The point of intersection represents the solution to the system.
Step 4: Verify the point by substituting the coordinates into both original equations. For example, if the lines intersect at (1, 3): plug into the first equation: 3 = 2(1)+1; plug into the second equation: 3 = 4(1)-1.
Final Answer: The solution to the system is the ordered pair (1, 3).

Solving a System by Graphing

QUESTION

Solve the system: 2x + y = 7 and x - 2y = 6 using substitution.

STEP-BY-STEP ANSWER:

Step 1: Solve the first equation for y: y = 7 - 2x.
Step 2: Substitute the expression for y into the second equation: x - 2(7 - 2x) = 6.
Step 3: Simplify and solve for x: x - 14 + 4x = 6, which gives 5x - 14 = 6, so 5x = 20 and x = 4.
Step 4: Substitute x = 4 back into y = 7 - 2x to find y: y = 7 - 8 = -1.
Final Answer: The solution is the ordered pair (4, -1).

Solving a System by Substitution

Scroll left
Scroll right

Common Mistakes

  • Misidentifying the slope and intercept when converting an equation to slope-intercept form.
  • Failing to check the solution by substituting the values back into the original equations.
  • Graphing inaccuracies due to scale misinterpretation which can lead to errors in finding the exact intersection point.
  • Confusing consistent systems (with at least one solution) with inconsistent systems (no solution) and not understanding the difference between independent and dependent equations.