Minnesota Mathematics Standards Standards Solving Equations of One Variable Stephanie Woldum, EDHD 5007

Objectives  To become familiar with solving equations of one variable.  To become comfortable with mathematical manipulations of equations.  To isolate the variable and to be able to find the number it represents.

Identifying the problem  We may only use the methods discussed here to solve equations of one variable. That is, equations that have only one type of variable.  Some examples….. x + 2 = 3x 3y + 7y = 10

Identifying the problem (cont)  We also want to apply the methods learned here to equations. Equations involve an equals sign.  Some examples… Expression Equation of one variable? Reason 3x + 2x =10 Yes = sign and only 1 variable 3x + 5z =11 No Involves more than 1 variable 11x - 2x No Doesn’t include an = sign 2x + 4y - 3z No No = sign, more than 1 variable

Links to more practice  If you would like more practice in identifying the types of problems we are concerned with click here. click here click here  If you feel comfortable with this concept and would like to move on, click here. click hereclick here

Identifying the problem: practice  Please indicate which expressions are equations of one variable. 1) 3x + 2x = 9 2) 11x + 3z = 2x 3)31,280x + 700x 4)11x + 2 = 13x Answers

Answers  Are the expressions equations of one variable? 1)Yes2)No3)No4)Yes

Isolating the variable  Once we have determined what type of problem we are dealing with we can select our method.  We want to solve the equation through the isolation of x on one side of the equation and all of the other numbers on the other side of the equation.

Method in Practice  To begin isolating x we need to combine all x factors. If factors are on the same side of the equation, you may just add or subtract these values from each other. Remember that we must abide by the Rules of Algebra. To see an example click here. here  You can request a copy of the Rules of Algebra by ing me here. here

Examples of combining x terms  2x + 3x = 5 is simplified to…  5x =5  Another example…  4x + 8 = 11x -4x -4x ~ ”whatever we do to one side of the equation we must do to the other.” -4x -4x ~ ”whatever we do to one side of the equation we must do to the other.” 8 = 7x 8 = 7x

Method (continued)  After we have combined all like terms, we need to start isolating the x term.  To do this we will work from the “outside in”. This means that we will start with the term farthest from the x term, usually some non-x term added to or subtracted from our x term. We will simplify through the use of “opposite operations”. “opposite operations”.“opposite operations”.  See some examples here. here

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

Examples of isolating x  3x + 12 = x = -4 3x = -4 Another example… 12x – 4 = x =12

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

The final step  2x = 8 divide by 2 both sides…. x = 4 x = 4 Another example… 3x = 15 divide both sides by 3… x = 5 x = 5

Summary  Once we have completely isolated x, we have finished the problem. We can check our solution by plugging in what we found x to be in the original equation. Then we need to make sure both sides of the equation are equivalent.  For more practice and explanation, try this online tutorial by clicking here. here

Minnesota Mathematics Standards  The following Minnesota Mathematics Standards were covered in this tutorial. Grade 9,10,11: I. Mathematical Reasoning. Standard: Apply skills of mathematical representation, communication and reasoning throughout the remaining three content strands. Benchmark IV. Grade 9,10,11: III. Patterns, Functions and Algebra. Sub-strand B. Standard: Solve simple equations and inequalities numerically, graphically, and symbolically. Use recursion to model and solve real-world and mathematical problems. Benchmarks I and VII. Back to title page. title

Credits  Picture credits (Slide 9 and Slide 16) © 2005 Microsoft Corporation This site can be viewed online at:  Online tutorial, ©2002 by Kim Peppard and Jennifer Puckett. The site can be viewed online at: hlab/int_algebra/int_alg_tut7_lineq.htm hlab/int_algebra/int_alg_tut7_lineq.htm hlab/int_algebra/int_alg_tut7_lineq.htm Back to title page. title