Minnesota Mathematics StandardsMinnesota Mathematics Standards

Solving Equations of One VariableSolving Equations of One Variable

Stephanie Woldum, EDHD 5007Stephanie Woldum, EDHD 5007

ObjectivesObjectives

To become familiar with solving equations of To become familiar with solving equations of one variable.one variable.

To become comfortable with mathematical To become comfortable with mathematical manipulations of equations.manipulations of equations.

To isolate the variable and to be able to find To isolate the variable and to be able to find the number it represents.the number it represents.

Identifying the problemIdentifying the problem

We may only use the methods discussed here We may only use the methods discussed here to solve equations of one variable. That is, to solve equations of one variable. That is, equations that have only one type of variable.equations that have only one type of variable.

Some examples…..Some examples…..

x + 2 = 3x x + 2 = 3x

3y + 7y = 103y + 7y = 10

Identifying the problem (cont)Identifying the problem (cont)

We also want to apply We also want to apply the methods learned the methods learned here to equations. here to equations. Equations involve an Equations involve an equals sign. equals sign.

Some examples…Some examples…ExpressionExpression Equation of one Equation of one

variable?variable?ReasonReason

3x + 2x =103x + 2x =10 YesYes = sign and only 1 = sign and only 1 variablevariable

3x + 5z =113x + 5z =11 NoNo Involves more than Involves more than 1 variable1 variable

11x - 2x11x - 2x NoNo Doesn’t include an Doesn’t include an = sign= sign

2x + 4y - 3z2x + 4y - 3z NoNo No = sign, more No = sign, more than 1 variablethan 1 variable

Links to more practiceLinks to more practice

If you would like more practice in identifying If you would like more practice in identifying the types of problems we are concerned with the types of problems we are concerned with click hereclick here..

If you feel comfortable with this concept and If you feel comfortable with this concept and would like to move on, would like to move on, click hereclick here..

Identifying the problem: practiceIdentifying the problem: practice

Please indicate which expressions are Please indicate which expressions are equationsequations of of one variable.one variable.

1) 3x + 2x = 91) 3x + 2x = 92) 11x + 3z = 2x2) 11x + 3z = 2x3)31,280x + 700x3)31,280x + 700x4)11x + 2 = 13x4)11x + 2 = 13x

AnswersAnswers

AnswersAnswers

Are the expressions equations of one variable?Are the expressions equations of one variable?

1)Yes1)Yes

2)No2)No

3)No3)No

4)Yes4)Yes

Isolating the variableIsolating the variable

Once we have determined what type of Once we have determined what type of problem we are dealing with we can select our problem we are dealing with we can select our method. method.

We want to solve the equation through the We want to solve the equation through the isolation of x on one side of the equation and isolation of x on one side of the equation and all of the other numbers on the other side of all of the other numbers on the other side of the equation.the equation.

Method in PracticeMethod in Practice

To begin isolating x we need to combine all x To begin isolating x we need to combine all x factors. If factors are on the same side of the factors. If factors are on the same side of the equation, you may just add or subtract these equation, you may just add or subtract these values from each other. Remember that we values from each other. Remember that we must abide by the Rules of Algebra. To see an must abide by the Rules of Algebra. To see an example click example click herehere..

You can request a copy of the Rules of You can request a copy of the Rules of Algebra by emailing me here.Algebra by emailing me here.

Examples of combining x termsExamples of combining x terms

2x + 3x = 5 is simplified to…2x + 3x = 5 is simplified to… 5x =55x =5

Another example…Another example… 4x + 8 = 11x4x + 8 = 11x -4x -4x -4x -4x ~ ”whatever we do to one side of the equation we must do to the other.”~ ”whatever we do to one side of the equation we must do to the other.”

8 = 7x8 = 7x

Method (continued)Method (continued)

After we have combined all like terms, we After we have combined all like terms, we need to start isolating the x term. need to start isolating the x term.

To do this we will work from the “outside in”. To do this we will work from the “outside in”. This means that we will start with the term This means that we will start with the term farthest from the x term, usually some non-x farthest from the x term, usually some non-x term added to or subtracted from our x term. term added to or subtracted from our x term. We will simplify through the use of We will simplify through the use of “opposite operations”.“opposite operations”.

See some examples See some examples herehere..

““Opposite Operations”Opposite Operations”

The concept of opposite operations involves using an The concept of opposite operations involves using an operation that will undo the equation and allow us to operation that will undo the equation and allow us to move terms around. If, for example, we want to move move terms around. If, for example, we want to move a +4 from one side of the equation to the other side, a +4 from one side of the equation to the other side, we use the opposite operation of addition; we subtract we use the opposite operation of addition; we subtract 4 from both sides. Similarly If we want to get rid of a 4 from both sides. Similarly If we want to get rid of a coefficient multiplied by an x term, we divide. coefficient multiplied by an x term, we divide.

To return to the “Start of Isolation” click To return to the “Start of Isolation” click herehere, to , to return to the “Completion of Isolation”, click return to the “Completion of Isolation”, click herehere..

Examples of isolating xExamples of isolating x

3x + 12 = 83x + 12 = 8 -12 -12-12 -12 3x = -43x = -4

Another example…Another example…12x – 4 = 812x – 4 = 8 +4 +4+4 +412x =1212x =12

Completing the IsolationCompleting the Isolation

Once we have all the regular numbers on one Once we have all the regular numbers on one side of the equation and all the x terms on the side of the equation and all the x terms on the other side we need to remove the coefficient in other side we need to remove the coefficient in front of the x. To do this we continue using front of the x. To do this we continue using “opposite operations.” “opposite operations.”

More examples follow on the next page.More examples follow on the next page.

The final stepThe final step

2x = 8 divide by 2 both sides….2x = 8 divide by 2 both sides….

x = 4x = 4

Another example…Another example…

3x = 15 divide both sides by 3…3x = 15 divide both sides by 3…

x = 5x = 5

SummarySummary

Once we have completely isolated x, we have Once we have completely isolated x, we have finished the problem. We can check our finished the problem. We can check our solution by plugging in what we found x to be solution by plugging in what we found x to be in the original equation. Then we need to make in the original equation. Then we need to make sure both sides of the equation are equivalent.sure both sides of the equation are equivalent.

For more practice and explanation, try this For more practice and explanation, try this online tutorial by clicking online tutorial by clicking herehere..

Minnesota Mathematics StandardsMinnesota Mathematics Standards

The following Minnesota Mathematics Standards were covered in this The following Minnesota Mathematics Standards were covered in this tutorial.tutorial.

Grade 9,10,11: I. Mathematical Reasoning.Grade 9,10,11: I. Mathematical Reasoning.

Standard: Apply skills of mathematical representation, communication and Standard: Apply skills of mathematical representation, communication and reasoning throughout the remaining three content strands. Benchmark IV.reasoning throughout the remaining three content strands. Benchmark IV.

Grade 9,10,11: III. Patterns, Functions and Algebra. Sub-strand B.Grade 9,10,11: III. Patterns, Functions and Algebra. Sub-strand B.

Standard: Solve simple equations and inequalities numerically, graphically, Standard: Solve simple equations and inequalities numerically, graphically, and symbolically. Use recursion to model and solve real-world and and symbolically. Use recursion to model and solve real-world and mathematical problems. Benchmarks I and VII.mathematical problems. Benchmarks I and VII.

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CreditsCredits

Picture credits (Slide 9 and Slide 16)Picture credits (Slide 9 and Slide 16)© 2005 Microsoft Corporation © 2005 Microsoft Corporation This site can be viewed online at: This site can be viewed online at: http://office.microsoft.com/clipart/default.aspx?lc=en-ushttp://office.microsoft.com/clipart/default.aspx?lc=en-us Online tutorial, Online tutorial, ©©2002 by Kim Peppard and Jennifer 2002 by Kim Peppard and Jennifer

Puckett. The site can be viewed online at: Puckett. The site can be viewed online at: http://www.wtamu.edu/academic/anns/mps/math/mathttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut7_lineq.htmhlab/int_algebra/int_alg_tut7_lineq.htm

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