Simultaneous Equations: Elimination, Substitution and Graphical method

CONTENT

• Solving Simultaneous Equations Using Elimination and Substitution Method
• Solving Equations Involving Fractions.
• Word problems.

SIMULTANEOUS LINEAR EQUATIONS

Methods of solving Simultaneous equation

i.          Elimination method

ii.          Substitution method

iii.         Graphical method

ELIMINATION METHOD

One of the unknowns with the same coefficient in the two equations is eliminated by subtracting or adding the two equations. Then the answer of the first unknown is substituted into either of the equations to get the second unknown.

SUBSTITUTION METHOD

One of the unknowns (preferably the one having 1 has its coefficient) is made the subject of the formula in one of the equations and substituted into the other equation to obtain the value of the first  unknown which is then substituted into either of the equations to get the second unknown.

FURTHER EXAMPLES

Solve for x and y simultaneously: 2x – 3y + 2 = x + 2y – 5 = 3x + y.

Solutions

            2x – 3y + 2 = x + 2y – 5 = 3x + y

            Form two equations out of the question

            2x – 3y + 2 = 3x + y

            x + 2y – 5 = 3x + y

                        OR

            2x – 3y + 2 = x + 2y – 5 ————- eq 1

            x + 2y – 5 = 3x + y       ————– eq 2

Rearrange the equations to put the unknown on one side and the constant at the other side.

            2x – 3y – x – 2y = – 5 – 2

            2x – x – 3y – 2y = -7

            x – 5y = -7 —————- eq 3

            From eqn 2

            x – 3x + 2y – y – 5

            – 2x + y = 5 ————- eq 4

Using substitution method solve eq 3 & 4

            x – 5y = -7 —————- eq 3

            -2x + y = 5 ————— eq 4

            Make y the subject in eq 4.

            y = 5 + 2x ————— eq 5

            Substitute eqn 5 into eqn 3.

            x – 5 (5 + 2x) = -7

            x – 25 – 10x = -7

            -9x – 25 = -7

            -9x = -7 + 25

            -9x = 18

                 x = 18/-9

X = -2

Substitute x = – 2 into eqn 5

y = 5 + 2x

y = 5 + 2(-2)

y = 5 – 4

y = 1

\ x = -2, y = 1

Simultaneous equations using Graphical method

EVALUATION

1.The  sum  of  two  numbers  is  110  and  their  difference  is  20. Find the two numbers.             

2.A pen  a  ruler  cost  #30.If  the  pen  costs  #8  more  than  the  ruler, how  much  does  each  item  cost ?

BONUS: 1 HOUR+ TONS OF SIMULTANEOUS EQUATION SOLVED EXAMPLES:

GENERAL EVALUATION AND REVISION   QUESTION

1. Solve the   following simultaneous equation:  3(2x – y) = x + y + 5 &  5(3x  –  2y) = 2 (x –y) + 1 

2. Five years ago, a father was 3 times as old as his son. Now, their combined ages amount to 110years. How old are they?

3. A  doctor  and  three  nurses  in  a  hospital  together  earn  #255 000 per  month, while  three  doctors  and  eight  nurses  together  earn  #720  000   per  month. Calculate (a) how much a doctor earns per month.  (b) How much  a  nurse  earns  per  month.

4.  Solve   simultaneously,   2^{x + 2y} = 1; 3^2x+y = 27

5.  Solve:  2x – 2y + 5 = 3x – 4y + 2 = -1                                                                      

WEEKEND ASSIGNMENT

1. If (x-y) log₁₀6  = log₁₀ 216 and 2 ^x+y =32 , calculate the values of x and y

a. x=1 , y=4        b. x= 4 , y =1       c. x=-4 , y= 1      d. x=4, y= -1

2. The point of intersection of the lines 3x- 2y =-12 and x + 2y = 4 is …

a. (5, 0)               b. (3, 4)               c. (-2, 5)              d. (-2, 3)

3. Find the value of (x – y), if 2x + 2y =16 and 8x – 2y = 44   a. 2    b. 4    c. 5     d. 6

4. If 5 ^{(p +2q)} =5 and 4 ^(p+3q) =16, the value of  3^(p+q)  is …..     a.0   b. -1   c.2    d. 1