Solving One Equation with One Variable

Two-step equations

Lesson Overview

Instructor Presentation

Introduction

A two-step equation is as straightforward as it sounds. You will need to perform two steps in order to solve the equation.

Goals

To have only variables on one side of the equal sign and numbers on the other side of the equal sign.

To have the number in front of the variable equal to one.

These equations can make use of any letter as a variable. The strategy for getting the variable by itself with a coefficient of 1 involves using opposite operations. For example, to move something that is added to the other side of the equation, you should subtract. The most important thing to remember in solving a linear equation is that whatever you do to one side of the equation, you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value from the other side.

Concept Exploration

Move the sliders and change the values for a, b and c in the equation.  You'll note:

Step 1:  Isolate the variable by adding or subtracting the constant on the SAME side of the variable to reduce the constant to 0.
Step 2:  Make the constant in front of the variable be 1 by dividing both sides by the constant in front of the variable.

(Note:  Remember to apply your change to both sides of the equation exactly - carry negative signs for example).

Manipulate[If[rn1==0,rn1=1,rn1];
Pane[
Text@Style[Grid[{
{Grid[{{Row[{ rn1,Style[" x ",Italic],If[rn2<0," - "," + "],Abs[rn2]," "}]," = ",Row[{" ",rn3,"     " }]
}}]},

{Style[If[rn2==0,"",
Grid[{{If[rn2<0,Row[{"+ ",Abs[rn2]}],Row[{"- ",Abs[rn2]}]], "   ",If[rn2<0,Row[{"+ ",Abs[rn2]}],Row[{"- ",Abs[rn2]}]]}
}]],Red]},

{Grid[{{Row[{If[rn1==1,"",rn1],Style[" x ",Italic]}]," = ",Row[{" ",rn3-rn2 }]
}}]},

{Style[If[rn1==0,"",
Grid[{{If[rn1<0,Row[{"/ ",Abs[rn1]}],Row[{"/ ",Abs[rn1]}]], "   ",If[rn1<0,Row[{"/ ",Abs[rn1]}],Row[{"/ ",Abs[rn1]}]]}
}]],Red]},

{If[rn1==1,"",
If[rn3-rn2==0,Grid[{{Style[" x ",Italic]," = ","0"}}],Grid[{{Style[ " x ",Italic], " = ",If[((rn3-rn2)/rn1)<0,Row[{"-",((ToString[Abs[rn3-rn2]])/(Abs[rn1]))}],Row[{((ToString[Abs[rn3-rn2]])/(Abs[rn1]))}]]}}]]]},

{If[rn1==1,"",
If[Abs[rn3-rn2]==Abs[Numerator[((rn3-rn2)/rn1)]],"",Grid[{{Style[" x ",Italic], " = ", ((rn3-rn2)/rn1)}}]]]}
},ItemSize->{12,1.5},Alignment->{Center,Center}, Dividers->{False,3->True}],20,Bold],{300,250},Alignment->{Center,Center}],
Style["a x + b = c",Bold],
{{rn1,2,"a"},-9,9,1,Appearance->"Labeled"},
{{rn2,-1,"b"},-9,9,1,Appearance->"Labeled"},
{{rn3,5,"c"},-20,20,1,Appearance->"Labeled"}
]

This came from the Wolfram Demonstrations Project, where you can download more mouse-driven examples for your class lessons: 
demonstrations.wolfram.com

Examples

Example 1

3 x + 7 = 5
      - 7    -7      (Step 1 - isolate the variable --- subtracted 7 from both sides)
__________
3x        = -2
----         ----
 3	  3     (Step 2 - make the constant in front of the variable be 1 --- divide both sides by 3)
 __________
 x = -(2/( 3))

Check Result

We can use Mathematica's Solve command to check our result.

(Note that we use 2 equal signs (==) when asking Mathematica to solve an equation.)

Click on Palettes ▶ Classroom Assistant for mouse-driven entry.  Maximize the Basic Commands section, click on y = x section, then click on Solve for a template.

Solve[lhs==rhs,var]

Fill in the template, then press the "Enter" key in "Classroom Assistant" or press "Shift" and "Enter" at the same time on your keyboard to evaluate.

Solve[3x+7==5,x]

We could also directly check our result:

3*-(2/3)+7

Example 2

-2t - 5  = -4
      +5    +5      (Step 1 - isolate the variable --- added 5 to both sides)
__________
-2t        =  1
----         ----
 -2	 -2     (Step 2 - make the constant in front of the variable be 1 --- divide both sides by -2)
 __________
 t = -(1/2)

Student Activity

Concept Exploration

Move the sliders in this example, what patterns do you see?

Manipulate[If[rn1==0,rn1=1,rn1];
Pane[
Text@Style[Grid[{
{Grid[{{Row[{ rn1,Style[" x ",Italic],If[rn2<0," - "," + "],Abs[rn2]," "}]," = ",Row[{" ",rn3,"     " }]
}}]},

{Style[If[rn2==0,"",
Grid[{{If[rn2<0,Row[{"+ ",Abs[rn2]}],Row[{"- ",Abs[rn2]}]], "   ",If[rn2<0,Row[{"+ ",Abs[rn2]}],Row[{"- ",Abs[rn2]}]]}
}]],Red]},

{Grid[{{Row[{If[rn1==1,"",rn1],Style[" x ",Italic]}]," = ",Row[{" ",rn3-rn2 }]
}}]},

{Style[If[rn1==0,"",
Grid[{{If[rn1<0,Row[{"/ ",Abs[rn1]}],Row[{"/ ",Abs[rn1]}]], "   ",If[rn1<0,Row[{"/ ",Abs[rn1]}],Row[{"/ ",Abs[rn1]}]]}
}]],Red]},

{If[rn1==1,"",
If[rn3-rn2==0,Grid[{{Style[" x ",Italic]," = ","0"}}],Grid[{{Style[ " x ",Italic], " = ",If[((rn3-rn2)/rn1)<0,Row[{"-",((ToString[Abs[rn3-rn2]])/(Abs[rn1]))}],Row[{((ToString[Abs[rn3-rn2]])/(Abs[rn1]))}]]}}]]]},

{If[rn1==1,"",
If[Abs[rn3-rn2]==Abs[Numerator[((rn3-rn2)/rn1)]],"",Grid[{{Style[" x ",Italic], " = ", ((rn3-rn2)/rn1)}}]]]}
},ItemSize->{12,1.5},Alignment->{Center,Center}, Dividers->{False,3->True}],20,Bold],{300,250},Alignment->{Center,Center}],
Style["a x + b = c",Bold],
{{rn1,2,"a"},-9,9,1,Appearance->"Labeled"},
{{rn2,-1,"b"},-9,9,1,Appearance->"Labeled"},
{{rn3,5,"c"},-20,20,1,Appearance->"Labeled"}
]

Skill Practice

Click on Palettes ▶ Classroom Assistant for mouse-driven entry.  Maximize the Basic Commands section, click on y = x section, then click on Solve for a template to check your work.

Easy

(1)  5 x + 1 = 4

x =

(2)  - 3 y + 5 = 8

y =

(3)  12 a + 7 = -9

a =

(4)   - 8 z - 4 = -8

z =

(5)   2 t - 5 = 6

t =

Difficult

(1)  (a/2)+5=11

a =

(2)  (1/3) (-2) b-8=10

b =

(3)  ((3 c)/5)+(1/2)=-(2/3)

c =

Exercises

(1) John wants to sell his comic books so he can purchase a video game.  He has $10 in his wallet and needs $55 for the game.  How much should he sell each of his 15 comic books for to ensure he makes just enough to purchase the game?

(2) Sue has 30 minutes for her lunch break.  If it takes her 10 minutes to drive to-and-from work to drive to the mall, how much time can she spend in each of the 2 stores she wants to visit during her lunch break?