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System of linear equationsSystem of linear equations can arise naturally from many real life examples.
I will only provide you with real life examples that lead to a system of linear equations and how to set up the system. Then, you will be ready to solve a system of equations using one of the three methods. Example #1 The sum of two numbers is twenty and their difference is ten. What are the two numbers? Here is how to set up the system: Let x be the first number Let y be the second number Then, x + y = 20 x − y = 10 You can also write: x + y = 20 y − x = 10 You will get the same answers except that the values for x and y will be swapped Example #2 You have 24 coins in your pockets that are worth 4.50 dollars. How many coins are quarters? How many coins are dimes? Here is how to set it up: Let q be the number of quarters Let d be the number of dimes Then, q + d = 24 25 × q + 10 × d = 450 The second equation is tricky. How did we get it? Since 1 quarter equal to 25 cents, q quarter equal to 25 × q If you had 6 quarters and you wanted to know how many cents are there for the 6 quarters, would you not do 6 × 25? Just say to yourself that now instead of 6 quarters you have q quarters. Does that make sense? In a similar way, since 1 dime equal to 10 cents, d dimes equal 10 × d What about the 450? 4.50 dollars times 100 = 450 cents Finally, since 25 × q represents how many cents you have for quarters and 10 × d represents how many cents you have for dimes, adding them should equal to the total of 450 cents Example #3 A cell phone plan offers 300 free minutes for a flat fee of 20 dollars. If your usage exceed 300 minutes, you pay 50 cents for each minute. A second cell phone plan offers 500 free minutes for a flat fee of 30 dollars. If your usage exceed 400 minutes, you pay 30 cents for each minute Model the cost of both plan with a system. Here is how to set up the system: Let x be the number of minutes you talk Let y be the cost y = 20 + 0.50 x y = 30 + 0.30 x Minus 0.50x from both sides in the first equation . Minus 0.30 x from both sides in the second equation We get: y − 0.50 x = 20 y − 0.30 x = 30 Now go to the lessons below to learn how to solve a system of linear equations. I have also included a system of linear equations solver Elimination method Substitution method System of linear equations solver |
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