Solving Equations:  Unknowns on Both Sides

Level 6

 

The objectives of this section are to:

*  solve equations with unknowns on each side, using the method of balancing;

*  solve number puzzles by first representing the puzzle as an equation.

 

 

Introduction

 

Consider the equation:    5t + 7 = 2t + 40.

 

In this equation, t is called an unknown.  We find the value of t by solving the equation. 

 

To solve the equation, we can use the method of balancing.  We try to simplify the equation by getting all the terms involving t on one side of the equation and all the number terms (without a t) on the other side.  Whatever operation we perform on one side of the equation, we must also do on the opposite side.

 

Example

 

Steps

Example

 

Solve    5t + 7 = 2t + 40

Step 1Simplify the equation by getting all t’s onto one side of the equation.  We want all the t’s on the side where there are most to start with.

       -2t                       -2t

 

            3t  +  7 =       40

Step 2Simplify the equation further by getting all the number terms onto the opposite side of the equation

               -7                   -7

 

            3t         =       33

Step 3Get the solution by dividing by the number in front of t.

        ÷3                         ÷3

 

 

             t        =        11

 

Note:  We can check whether the solution is correct by substituting the value of t back into both sides of the original equation:

            The left hand side is  5t + 7 = 55 + 7 = 62

            The right hand side is  2t + 40 = 22 + 40 = 62

 

 

Equations with brackets

 

If the equation contains any brackets, we begin by expanding the brackets:

 

 

Example

Solve                3x + 19 = 4(2x + 1) + x

 

First step:  Expand out the bracket on the right hand side (by multiplying everything in the bracket by 4)

3x + 19 = 8x + 4 + x

Second step:  Simplify the right hand side by collecting like-terms together.

3x + 19 = 9x + 4

Third step:  Get all the x’s together on the right hand side (that is where there are most to start with).  We do this by subtracting 3x from both sides.

        19 = 6x + 4  

Fourth step:  Get all the number terms on the opposite side.  We do this by subtracting 4 from both sides.

         15 = 6x

Fifth step:  Get the solution by dividing both sides by 6 (i..e the number in front of x).

          2.5 = x

 

So the solution to this equation is   x = 2.5

 

 

Equations with negative terms

 

Example:

Solve    4y – 6 = 18 – 6y

 

Steps

Example

 

Solve    4y - 6 = 18 – 6y

Step 1Simplify the equation by getting all y’s onto one side of the equation.  In this case, there are 4y on the left and -6y on the right. So, taking into account the signs, the left side has more.  To remove the -6y from the right hand side, we ADD 6y to both sides.

     + 6y                        + 6y

 

            10y - 618

Step 2:  Next we try to get the numbers on the opposite side.  Here we remove the -6 by ADDING 6 to both sides.

             + 6               + 6

 

            10y       =  24

Step 3Get the solution by dividing by the number in front of y.

      ÷10                         ÷10

 

 

               y        =        2.4

 

So the solution to the equation is y = 2.4

 

Example 2:

Solve 40 – 5n = 28 – 3n

 

Steps

Example

 

Solve    40 – 5n = 28 – 3n

Step 1Simplify the equation by getting all n’s onto one side of the equation.  In this case, there are -5n on the left and -3n on the right. So, taking into account the signs, the right hand side has more (-3 is larger than -5).

           + 5n                     + 5n

 

             40        = 28  + 2n

Step 2Next we try to get the numbers on the opposite side.  Here we remove the 28 by SUBTRACTING 28 from both sides.

      -28                     -28

 

            12       =        2n

Step 3Get the solution by dividing by the number in front of n.

      ÷2                         ÷2

 

 

             6       =        n

 

So, the solution to the equation is n = 6.

 

 

Example 3:

Solve 7x – 2 = 2x – 9

 

Steps

Example

 

Solve    7x – 2 = 2x – 9

Step 1Simplify the equation by getting all x’s onto one side of the equation.  In this case, we are looking for the x’s to appear on the left side (as there are more x’s there to begin with).

 

        - 2x                - 2x

 

             5x  - 2  =      -9  

Step 2Next we try to get the numbers on the opposite side.  Here we remove the -2 by ADDING 2 to both sides.

              +2                  +2

 

              5x      =        -7

Step 3Get the solution by dividing by the number in front of x.

      ÷5                         ÷5

 

 

               x       =        -1.4

 

So, the solution to the equation is x = -1.4.

 

 

 

Solving puzzles

 

Introduction: 

 

Consider this number puzzle.

 

            Sally thinks of a number.

            She multiplies it by 7.

            She adds on 12.

            Her answer is 11 times the number she started with.

 

            Work out the number Sally thought of.

 

 

We can solve this number puzzle by first writing it as an equation.

 

Number puzzle

Algebra

Sally thinks of a number.

Let the number she thinks of be n.

She multiplies it by 7.                           

She is now thinking of 7n.

She adds on 12.

She is now thinking of 7n + 12.

Her answer is 11 times the number she started with.

We can form an equation:

            7n + 12 = 11n

Work out the number Sally thought of.

We now solve the equation:

                     12 = 4n     (subtracting 7n)

                        3 = n      (dividing by 4)

 

So Sally must have thought of the number 3.  We can check the answer by working through the number puzzle using 3 as Sally’s number.

 

 

Example:

 

            Henri thinks of a number.

           He adds on 4.

           He multiplies it by 5.                                 

           His answer is 9 times the number he started with.

 

           Work out the number Henri thought of.

 

 

To find Henri’s starting number, we use the same method as above:

 

Number puzzle

Algebra

Henri thinks of a number.

Let the number he thinks of be n.

He adds on 4.

He is now thinking of n + 4.

He multiplies it by 5.

He is now thinking of 5(n + 4).

This is equivalent to  5n + 20.

His answer is 9 times the number he started with.

We can form an equation:

            5n + 20 = 9n

 

We now solve the equation:

                     20 = 4n     (subtracting 5n)

                        5 = n      (dividing by 4)

 

So Henri must have thought of the number 5.