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  1. 1. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1Solving simultaneous equationsThe terms simultaneous equations and systems of equations refer to conditionswhere two or more unknownvariables are related to each other through an equalnumber of equations. Consider the following example:For this set of equations, there is but a single combination of values for x and y that willsatisfy both. Eitherequation, considered separately, has an infinitude ofvalid (x,y) solutions, but together there is only one. Plotted on a graph, this conditionbecomes obvious:Each line is actually a continuum of points representing possible x and y solution pairsfor each equation. Each equation, separately, has an infinite number of ordered pair(x,y) solutions. There is only one point where the two linear functions x + y = 24 and 2x -y = -6 intersect (where one of their many independent solutions happen to work for bothequations), and that is where x is equal to a value of 6 and y is equal to a value of 18.Usually, though, graphing is not a very efficient way to determinethe simultaneous solution set for two or more equations. It is especially impractical for
  2. 2. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1systems of three or more variables. In a three-variable system, for example, the solutionwould be found by the point intersection of three planes in a three-dimensionalcoordinate space -- not an easy scenario to visualize.Substitution methodSeveral algebraic techniques exist to solve simultaneous equations. Perhaps theeasiest to comprehend is the substitution method. Take, for instance, our two-variableexample problem:In the substitution method, we manipulate one of the equations such that one variable isdefined in terms of the other:Then, we take this new definition of one variable and substitute it for the samevariable in the otherequation. In this case, we take the definition of y, which is 24 - x andsubstitute this for the y term found inthe other equation:Now that we have an equation with just a single variable (x), we can solve it using"normal" algebraic techniques:
  3. 3. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1Now that x is known, we can plug this value into any of the original equations and obtaina value for y. Or, to save us some work, we can plug this value (6) into the equation wejust generated to define y in terms ofx, being that it is already in a form to solve for y:Applying the substitution method to systems of three or more variables involves asimilar pattern, only with more work involved. This is generally true for any method ofsolution: the number of steps required for obtaining solutions increases rapidly witheach additional variable in the system.
  4. 4. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1To solve for three unknown variables, we need at least three equations. Consider thisexample:Being that the first equation has the simplest coefficients (1, -1, and 1, for x, y, and z,respectively), it seems logical to use it to develop a definition of one variable in terms ofthe other two. In this example, Ill solve for x in terms of y and z:Now, we can substitute this definition of x where x appears in the other two equations:Reducing these two equations to their simplest forms:
  5. 5. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1So far, our efforts have reduced the system from three variables in three equationsto two variables in twoequations. Now, we can apply the substitution technique again tothe two equations 4y - z = 4 and -3y + 4z = 36 to solve for either y or z. First, Illmanipulate the first equation to define z in terms of y:Next, well substitute this definition of z in terms of y where we see z in theother equation:
  6. 6. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1Now that y is a known value, we can plug it into the equation defining z in terms of y andobtain a figure forz:Now, with values for y and z known, we can plug these into the equation where wedefined x in terms of yand z, to obtain a value for x:
  7. 7. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1SIMULTANEOUS EQUATION BY ELIMINATIONTheory: In the ‘elimination’ method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found. To add two equations, add the left hand expressions and right hand expressions separately. Similarly, to subtract two equations, subtract the left hand expressions from each other, and subtract the right hand expressions from each other. The following examples will make this clear. Example 1: Consider these equations: 2x 5y = 1 3x + 5y = 14 The first equation contains a ‘ 5y’ term, while the second equation contains a ‘+5y’ term. These two terms will cancel if added together, so we will add the equations to eliminate ‘y’. To add the equations, add the left side expressions and the right side expressions separately.
  8. 8. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 2x 5y = 1 + 3x + 5y = +14 (2x 5y) + (3x + 5y) = 1 + 14 Simplifying, 5y and +5y cancel out, so we have: 5x = 15 Therefore ‘x’ is 3. By substituting 3 for ‘x’ into either of the two original equations we can find ‘y’. Example 2: The elimination method will only work if you can eliminate one of the variables by adding or subtracting the equations as in example 1 above. But for many simultaneous equations, this is not the case. For example, consider these equations: 2x + 3y = 4 x 2y = 5 Adding or subtracting these equations will not cancel out the ‘x’ or ‘y’ terms. Before using the elimination method you may have to multiply every term of one or both of the equations by some number so that equal terms can be eliminated. We could eliminate ‘x’ for this example if the second equation had a ‘2x’ term instead of an ‘x’ term. By multiplying every term in the second equation by 2, the ‘x’ term will become ‘2x’, like this:
  9. 9. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 x 2 2y 2 = 5 2 giving: 2x 4y = 10 Now the ‘x’ term in each equation is the same, and the equations can be subtracted to eliminate ‘x’: 2x + 3y = 4 2x 4y = 10 (2x + 3y) (2x 4y) = 4 10 Removing the brackets and simplifying, the ‘2x’ terms cancel out, so we have: 7y = 14 So y=2 The other variable, ‘x’, can now be found by substituting 2 for ‘y’ into either of the original equations.Sometimes both equations must be modified in order to cancel a variable. Forexample, to cancel the ‘y’ terms for this example, we could multiply the firstequation by 4, and the second equation by 3. Then there would be a ‘12y’ termin the first equation and a ‘-12y’ term in the second equation. Adding theequations would then eliminate ‘y’.