Solving

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27. Aug 2012
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Solving

• 1. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 Solving simultaneous equations The terms simultaneous equations and systems of equations refer to conditions where two or more unknownvariables are related to each other through an equal number of equations. Consider the following example: For this set of equations, there is but a single combination of values for x and y that will satisfy both. Eitherequation, considered separately, has an infinitude of valid (x,y) solutions, but together there is only one. Plotted on a graph, this condition becomes obvious: Each line is actually a continuum of points representing possible x and y solution pairs for 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 both equations), 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 determine the simultaneous solution set for two or more equations. It is especially impractical for
• 2. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 systems of three or more variables. In a three-variable system, for example, the solution would be found by the point intersection of three planes in a three-dimensional coordinate space -- not an easy scenario to visualize. Substitution method Several algebraic techniques exist to solve simultaneous equations. Perhaps the easiest to comprehend is the substitution method. Take, for instance, our two-variable example problem: In the substitution method, we manipulate one of the equations such that one variable is defined in terms of the other: Then, we take this new definition of one variable and substitute it for the same variable in the otherequation. In this case, we take the definition of y, which is 24 - x and substitute 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. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 Now that x is known, we can plug this value into any of the original equations and obtain a value for y. Or, to save us some work, we can plug this value (6) into the equation we just 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 a similar pattern, only with more work involved. This is generally true for any method of solution: the number of steps required for obtaining solutions increases rapidly with each additional variable in the system.
• 4. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 To solve for three unknown variables, we need at least three equations. Consider this example: 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 of the other two. In this example, I'll 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. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 So far, our efforts have reduced the system from three variables in three equations to two variables in twoequations. Now, we can apply the substitution technique again to the two equations 4y - z = 4 and -3y + 4z = 36 to solve for either y or z. First, I'll manipulate the first equation to define z in terms of y: Next, we'll substitute this definition of z in terms of y where we see z in the other equation:
• 6. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 Now that y is a known value, we can plug it into the equation defining z in terms of y and obtain a figure forz: Now, with values for y and z known, we can plug these into the equation where we defined x in terms of yand z, to obtain a value for x:
• 7. Name: Ma. Angeli Oguan BSHM1H Subject: MathOne1 SIMULTANEOUS EQUATION BY ELIMINATION Theory: 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. 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. 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. For example, to cancel the ‘y’ terms for this example, we could multiply the first equation by 4, and the second equation by 3. Then there would be a ‘12y’ term in the first equation and a ‘-12y’ term in the second equation. Adding the equations would then eliminate ‘y’.