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Solving Systems of Equations by Graphing: Method, Applications, and Examples, Summaries of Algebra

McKendree University (McK)Algebra

This section covers the method of solving systems of linear equations by graphing, its applications, and examples. It includes the concept of a system of linear equations, the graphical solution process, and the interpretation of the results. The section also provides an example of a break-even point using a cost and revenue system.

What you will learn

  • What is the graphical method for solving systems of linear equations?
  • How do you check a proposed solution using the graphing method?
  • What are the different types of solutions for a system of linear equations?

Typology: Summaries

2021/2022

Uploaded on 09/12/2022

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Section 3.1
Solving Systems of Equations by Graphing
What is a System of Equations?
Solving Linear Systems – The Graphing Method
Consistent Systems – one point (x,y) solution
Inconsistent Systems – no solution
Dependant Systems – infinite solutions
Solving Equations Graphically
Concept:
A System of Linear Equations
Any pair of Linear Equations can be a System
A Solution Point is an ordered pair (x,y) whose
values make both equations true
When plotted on the same graph, the solution is
the point where the lines cross (intersection)
Some systems do not have a solution
Why Study Systems of Equations?
We will study systems of 2 equations in 2 un knowns (usually x and y)
The algebraic methods we use to solve them will also be useful in
higher degree systems that involve quadratic equations or systems of 3
equations in 3 unknowns
pf3
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Download Solving Systems of Equations by Graphing: Method, Applications, and Examples and more Summaries Algebra in PDF only on Docsity!

Section 3.

Solving Systems of Equations by Graphing

 What is a System of Equations?

 Solving Linear Systems – The Graphing Method

 Consistent Systems – one point ( x,y ) solution

 Inconsistent Systems – no solution

 Dependant Systems – infinite solutions

 Solving Equations Graphically

Concept:

A System of Linear Equations

 Any pair of Linear Equations can be a System

 A Solution Point is an ordered pair (x,y) whose

values make both equations true

 When plotted on the same graph, the solution is

the point where the lines cross (intersection)

 Some systems do not have a solution

Why Study Systems of Equations?

We will study systems of 2 equations in 2 unknowns (usually x and y) The algebraic methods we use to solve them will also be useful in higher degree systems that involve quadratic equations or systems of 3 equations in 3 unknowns

A “Break Even Point” Example

A $50 skateboard costs $12.50 to build,

once $15,000 is spent to set up the factory:

 Let x = the number of skateboards

 f(x) = 15000 + 12.5x (total cost equation)

 g(x) = 50x (total revenue equation)

Using Algebra to

Check a Proposed Solution

Is (3,0) also a solution?

Estimating a Solution using

The Graphing Method

 Graph both equations on the same graph paper

 If the lines do not intersect, there is no solution

 If they intersect:

 Estimate the coordinates of the intersection point  Substitute the x and y values from the (x,y) point into both original equations to see if they remain true equations

Practice – Solving by Graphing

Consistent: infinite sol’s

3y – 2x = 6  (0,2) and (-3,0) -12y + 8x = -24  (0,2) and (-3,0) Looks like a dependant system … Check it: divide all terms in the 2nd^ equation by - and it becomes identical to the 1st^ equation therefore, consistent, dependant system



Solving Equations by Graphing

 Equations in one unknown can be split into

two linear equations:

 2x + 4 = -2 

 f(x) = 2x + 4 and g(x) = -

 When the two linear equations are graphed as

a system, the solution is the point (-3,-2)

 The x-coordinate is the solution to the original

equation in one unknown!

 The lines above cross at (-3,-2)  x = -

The Downside of Solving by Graphing:

It is not Precise

Summary of Section 3.

 Solve Systems by Graphing Them Together

 Graph neatly both lines using x & y intercepts

 Solution = Point of Intersection (2 Straight Lines)

 Check by substituting the solution into all equations

 Cost and Revenue lines cross at “Break Even Point”

 A Consistent System has one solution (x,y)

 An Inconsistent System has no solution

The lines are Parallel (have same slope, different y-intercept)

 A Dependent System happens when both equations

have the same graph (the lines have same slope and y-intercept)

 Graphing can solve equations having one variable

What Next?

 Present Section 3.

Solving Systems Algebraically