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**Revision Simultaneous Equations I**

Solving equations simultaneously. By I Porter.

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**Graphical Method This is NOT the best or recommended method.**

The graphical method of solving simultaneous equations involves these steps Step 1: Complete a table of values for each equation. Step 2: Draw the graphs of both equations on the same number plane. Step 3: From the graphs, find the point of intersection. Step 4: Use the point of intersection to write the solution. Solve the simultaneous equations x + y = 6 and 2x - y = 3 Table of values: x + y = 6 2x - y = 3 x 1 2 3 y 5 4 Point of intersection (3,3). Solution is x = 3 and y = 3. x 1 2 3 y -1

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Elimination Method. The elimination method is an algebraic technique that gives the exact solution of simultaneous equations. In this method, the aim is to eliminate one of the variables so that the value of the other variable can be found. The step to follow are: Step 1: Make sure that the coefficients of one of the variables are the same. Solve simultaneously 2x - 5y = 13 5x - 3y = -15 To eliminate y, multiply (1) by 3 & multiply (2) by 5 < (1) < (2) Step 2: Eliminate one variable by adding or subtracting the pair of equations. So 6x - 15y = 39 25x - 15y = -75 To eliminate y, Subtract the two equation as they both contain -15y, same sign (-). (3) - (4) < (3) < (4) Step 3: Solve the new equation to find the value of the remaining variable. -19x = 114 Solve for x. x = -6 Step 4: Substitute this value into one of the original equations to find the value of the second variable. Hence, 2(-6) - 5y = 13 Substitute x = -6 into (1) to find y y = 13 The solution is x = -6 and y = -5. Step 5: Write your answer, showing the values both variables clearly. - 5y = 25 y = 5

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**Example 1: Solve the simultaneous equations, 2x + 3y = -8 and 4x - y = 12**

< (1) To eliminate y, multiply (1) by the absolute coefficient of y in (2), 1 Multiply (2) by the absolute coefficient of y in (1) , 3 3 X 4x - y = 12 < (2) So, 2x + 3y = -8 12x - 3y = 36 < (3) To eliminate y, Add the two equation as they have opposite in sign (3) + (4) < (4) 14x = 28 Solve for x. x = 2 Substitute x = 2 into (1) to find y Hence, 2(2) + 3y = -8 4 + 3y = -8 3y = -12 y = -4 Remember If the variable you are eliminating have the Same signs => subtract Different signs => add two equations. The solution is x = 2 and y = -4.

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**Example 2: Solve the simultaneous equations, 5x + 3y = -19 and 2x + 4y = -16**

< (1) To eliminate y, multiply (1) by the absolute coefficient of y in (2), 4 Multiply (2) by the absolute coefficient of y in (1) , 3 3 X 2x + 4y = -16 < (2) So, 20x + 12y = -76 6x + 12y = -48 < (3) To eliminate y, Subtract the two equation as the same sign (3) - (4) < (4) 14x = -28 Solve for x. x = -2 Substitute x = -2 into (1) to find y Remember If the variable you are eliminating have the Same signs => subtract Different signs => add two equations. Hence, 5(-2) + 3y = -19 y = -19 3y = -9 y = -3 The solution is x = -2 and y = -3.

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**Exercise: Solve the following simultaneous equations using the elimination method.**

a) 3a + b = 4 & 4a - b = 10 Ans: a = 2, b = -2 b) 2a - 7b = 19 & a + 2b = 4 Ans: a = 6, b = -1 c) 4a + 5b = 22 & a + b = 10 Ans: a = 28, b = -18 d) 5a - 3b = -1 & 8a - 2b = 4 Ans: a = 1, b = 2 e) 3a - 4b = 24 & 4a - 2b = 12 Ans: a = 0, b = -6

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5.1 Solving Systems of Linear Equations by Graphing

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