Download Algebra 1: Unit 5: Systems of Equations and

Algebra 1: Unit 5: Systems of Equations and Inequalities
Chapter Overview
Lesson 5.1: Solving Systems of Equations Using Graphs: ....................................................................................................................... 2
Lesson 5.2: Solving Systems of Equations Using Substitution: ............................................................................................................ 5
Lesson 5.3: Solving Systems of Equations Using Elimination ............................................................................................................... 6
Lesson 5.4: Inequalities in One Variable........................................................................................................................................................ 9
Lesson 5.5: Graphing Inequalities in Two Variables...............................................................................................................................11
Lesson 5.6: Systems of Inequalities ...............................................................................................................................................................12
Colorado Mathematics Standards

2 Students use algebraic methods to explore, model, and describe patterns and functions involving
numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning
used in solving these problems.
o 2.1 model real-world phenomena (for example, distance versus-time relationships,
compound interest, amortization tables, mortality rates) using functions, equations,
inequalities, and matrices
o 2.3 solve problems involving functional relationships using graphing calculators and/or
computers as well as appropriate paper-and-pencil techniques
Diploma Plus Competencies
Diploma Plus Core Subject Competencies: Math
» M01 Problem Solving: Solve mathematical problems.
Diploma Plus Subject Competencies: Personal
» P01 Self-management: Take responsibility for changing personal behaviors or acquiring skills that lead
to both social and academic success.
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Lesson 5.1: Solving Systems of Equations Using Graphs:
Objective: You will need to be able to model real world situations with systems of two equations; solve systems of two
linear equations using tables and graphs; and understand that the intersection of two lines provides a solution to the
system and thus to a real-world problem.
A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system
of equations, we try to find where the two equations will intersect.
When we solve systems of equations by graphing, we will:
1) Put both equations in slope intercept form
2) Graph both lines on a coordinate plane
3) Label where they intersect.
Quick Review:
Write down what you remember about slope intercept form. Remember, it is AWESOME!
Practice graphing these lines..If you are stuck, STOP and ask for a mini lesson.
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Practice:
Now that you are comfortable graphing lines, lets try solving systems of equations by graphing. Try to not let the
long name fool you; all you are doing is graphing two lines.
Easy, right? Now, write the solution, or the point of intersection, as an ordered pair. If you lines crossed when x=3
and y= -2 then your ordered pair would look like (3, -2).
Learn how to solve these on your calculator. I promise, it’s easy!!!
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Word Problems
Let’s try and solve systems of equations in word problems.
Here is an example:
Work through the exercises on pages 276-279. Focus on #6, 7, and 8 on page 278. Pick one of the problems to
graph by hand. Graph the other two on your calculator. Attach your work to the back.
Questions:
1. Why do you think there is only one solution for a system of equations?
2. What does it mean if you are unable to find a solution?
3. How can you check your work to make sure your solution is accurate?
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Lesson 5.2: Solving Systems of Equations Using Substitution:
Objective: You willneed to understand the limitations of solving systems of equations graphically, and learn how to
solve them using substitution.
The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of
the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the
chosen variable and solving for the other. Then you back-solve for the first variable. Here is how it works.
STEPS:
1)
2)
3)
4)
5)
Solve for 1 variable
Substitute that variable into your other equation.
Solve for the second variable.
Plug in what you just solved for in either equation.
Solve for remaining variable.
Solve the following system by substitution.
2x – 3y = –2
4x+ y = 24
The idea here is to solve one of the equations for one of the variables, and plug this into the other equation. It does not
matter which equation or which variable you pick. There is no right or wrong choice; the answer will be the same,
regardless. But — some choices may be better than others.
For instance, in this case, can you see that it would probably be simplest to solve the second equation for "y =", since
there is already a y floating around loose in the middle there? I could solve the first equation for either variable, but I'd
get fractions, and solving the second equation for x would also give me fractions. It wouldn't be "wrong" to make a
different choice, but it would probably be more difficult. Being lazy, I'll solve the second equation for y:
4x + y = 24
y = –4x + 24
Now I'll plug this in ("substitute it") for "y" in the first equation, and solve for x:
2x – 3(–4x + 24) = –2
2x + 12x – 72 = –2
14x = 70
x = 5Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
Now I can plug this x-value back into either equation, and solve for y. But since I already have an expression for "y =",
it will be simplest to just plug into this:
y = –4(5) + 24 = –20 + 24 = 4
Then the solution is (x, y) = (5, 4).
Read the examples on page 281 and 283 – 284 for additional support.
You can also watch the video here: http://www.youtube.com/watch?v=JYqBzPK5qfU for a verbal walk through.
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Practice:
Attach your work to the back.
Questions:
3. What does substitution mean?
4. When would you use this method?
Lesson 5.3: Solving Systems of Equations Using Elimination
Objective: In this lesson you will need to know how to solve systems of equations using elimination, and continue to
model real-world situations with two linear equations.
Watch the video here: http://www.brightstorm.com/math/algebra/solving-systems-of-equations/solvingsystems-of-equations-using-elimination
Practice:
Read through both Examples (skip the Investigation) and do the problems below.
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Also solve 13 and 14 on page 294 and attach to the back.
Questions:
Fill in the pros/cons T-chart about the elimination method.
Pros Cons
Review:
Guidelines for Solving a System of Linear Equations
To help you decide which method (graphing, substitution, or elimination) to use solving a system of linear
equations, consider the following.
 The graphing method is useful for approximating a solution, checking a solution and for providing a visual
model for the problem.

To find an exact solution, use either substitution or elimination.
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
Remember that parallel lines have no solutions and two different forms of the same equation will represent
the same line and have infinite solutions.

For linear systems in which one variable has a coefficient of 1 or -1, substitution may be more efficient.

In other cases the linear combination (elimination) method is usually more efficient.
Questions:
Which method do you prefer and why?
Graphing
Substitution
When would you use
it?
How do you use it?
What is real world scenario that involves a system of equations?
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Elimination
Lesson 5.4: Inequalities in One Variable
Objective:Here you will learn to write and solve one-variable inequalities, graph solutions to one-variable inequalities
on a number line, write inequality sentences based on graphed intervals, and learn how to change signs when
multiplying or dividing both sides of one-variable inequalities by negative numbers.
Inequality Signs:
= means “is equal to”
>means "is greater than"
<means "is less than"
≥ means "is greater than or equal to"
≤ means "is less than or equal to"
Whenever you are asked to solve inequalities you should always think of the inequality as an equal sign.
Think to yourself, “How would I solve this if it were an equals sign?” The same rules apply.
The only new rule is: If you have to multiply or divide the inequality by a negative number, you
must flip the sign!
Ex. < becomes > and vice versa
Ex. > becomes < and vice versa.
Quick Review:
Try solving each of these for the variable.
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Practice Graphing:
Here is some practice graphing inequalities. I always read the variable first (you may need to change the order of
the last few) and put a circle at the number.

If the inequality reads “greater than” or “less than”, I leave the circle open (or unshaded) because the
variable cannot equal that number.

If the inequality reads “greater than or equal to” or “less than or equal to”, I fill in the circle because the
variable CAN equal that number.
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Final practice for this section:
Lesson 5.5: Graphing Inequalities in Two Variables.
Objective: Be able to solve two-variable inequalities for y; graph inequalities on the coordinate plane and show
solutions as the intersection of two half planes; and interpret graphs of half planes and write corresponding
inequalities.
This is VERY similar to graphing a line with two variables.
1. Put your line in Slope Intercept Form
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2. Graph it. Draw a dotted line if the inequality is “greater than” or “less than”. Draw a straight line if the
inequality is “greater than or equal to” or “less than or equal to”.
3. Shade either above or below the line depending on what the inequality says.
Practice
Work through Example A and B for more in-depth understanding. Do problems 1, 2, 6, and 7 on pages 316 & 317.
Attach to back.
Questions:
What is a real world situation where it would be important to know how to graph inequalities?
Lesson 5.6: Systems of Inequalities
Objective:In this lesson you will learn how to solve systems of inequalities by graphing, interpret mathematical
solutions in terms of the problem’s context, and write inequalities to represent constraints in application problems.
This is the big kahuna. This is where all of your Unit 6 Knowledge comes together!!! Solving systems of
inequalities is essentially the thing as systems of equations. The only difference is that you have to shade
regions. Your solutions will be areas that have been shaded twice!
Practice:
Do problems 1, 2, and 5 on pages 324 and 325.
Questions:
How is solving systems of inequalities different than solving systems of equations?
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