Why study mathematics? - Solution of linear by linear combinations

Now that we have seen how to solve a system of linear equations using the substitution method, you move to a more convenient method known as linear combinations. With this method, in addition, subtraction reputation, it eliminates a variable from a most appropriate one of equations. We can eliminate a variable and solve for others. Once done, you use the other equation to solve for the other variables.

This method can be done algorithmically and thenThese are the steps for solving a system of linear combinations:

Step 1: Arrange the equations with like terms in columns.

Step 2: Multiply one or both of the coefficients of the equation in order to dilute the variables are opposites of one of the.

Step 3: Add the equations of the previous step. Combine like terms eliminated one of the variables and fixes to others.

Step 4: Replace the value of the resultantto solve the equations in a previous step and the other variable.

Step 5: Check the solution in each of the original equations.

To illustrate the algorithm, we solve the following system: 4x + 3y = 16 and 2x - 3y = 8 Initially, these two equations is in columns, so that variables such as line-up. So we

4x + 3y = 16

2x - 3y = 8

As you can see that the coefficients of the y-contrast conditions, there is no need to multiply the equationsGet this form. And so with only two, disappear so that the terms-y. We 6x = 24 Solving for x we have x = 4 Substituting this value in the first equation, for example, gives four (4) + 3y = 16 or 16 + 3y = 16 3y = 0 or y = 0. Check with the inclusion of these values generated in each original equation a true statement. Thus, the solution is x = 4 y = 0 or the point (4, 0) as the intersection of these two lines in a coordinate system.

Let us see howthe method of linear combinations of a model of historical problem. According to legend, the famous greek mathematician Archimedes, the ratio between the weight of an object and its volume is used to determine that there is fraud in the manufacture of a golden crown. The way this was done by applying the principle of displacement volume. You see, when a crown of pure gold, so the same volume the same amount to replace the gold. The number that follows, the concept ofThe density is also used. By definition, the density of an object equals its mass divided by volume. Gold has a density of 19 grams per cubic centimeter. Silver has a density of 10.5 grams per cubic centimeter. We are made so that the problem follows to use.

Problem: Suppose a crown of gold and silver with some suspicion, he weighed 714 grams and had a volume of 46 cubic centimeters. What percentage of the crown was silver?

To resolve this problem, we observe thatthe volume of the volume of gold silver, more must be equal if the total volume of 46. Since they are the densities of gold and silver, white, and we know that the density of times the corresponding density, we have that golden moment of the density, the volume of gold, more silver is the silver density times band the total weight. We let G = Gold S = volume of silver and volume. Now we can translate the problem in mathematics and a linear system.

We have a G + S + G = 46 and 19 = 10.5S 714th Puttingcolumns in these equations, we

G + S = 46

19G + = 10.5S 714th

Now multiply the first equation of -19 to opposite coefficients for G. So we get

-19G +-19S = -874

19G + = 10.5S 714th

Adding the two equations we have - 8.5S = -160. Dividing both sides by -8.5, we have S = 18.8, we set up in 19. The volume of silver is 19 cubic centimeters, and the percentage of silver in the crown is 19/46 or 41% for the nearest percentage.Remember, this method next time someone tries to pawn on you from pure gold, when in fact the reality is quite different. Watch out for Fool's Gold 's!

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5 Strategies to Teach Your Children About Saving Money

Teaching your children financial skills is critical for their future. 80% of parents believe that their children are being taught personal money matters in school, yet 90% of high school students and 87% of college students say that whatever they know about money they learn from their parents. Among parents with children 5 and older, only 26% feel well enough prepared to teach their kids about personal finances. Jump$tart Coalition for Personal Financial Literacy measured 12th graders' knowledge of personal finance basics and found that only 10% of high school graduates could satisfactorily answer questions about personal finance.

Not sure where to start in talking to you kids about money? You're not alone. But, much like teaching your kids to look both ways before crossing the street, managing money, is a parental responsibility that safeguards kids' future. Good habits start early in life and the savings habit brings lifelong benefits. Kids are interested in money and they can learn by example and by doing. Sharing how and why your family is saving emphasizes the importance of this positive, lifelong habit.

Engage your children using some of these simple, fun suggestions and help them learn the value of money:

1. Explain to your kids what money is all about.
You can start doing this once you see that your kids are already able to learn how to count. The earlier you can teach a child or teenager about money, including earning money, saving money, and spending money responsibility, the better prepared they will be to manage their own money.

2. Talk to your child about the family budget.
Allow them to ask questions about household finances and how you manage the household budget. Reinforce the learning process by budgeting for a family outing or a purchase.

3. Encourage children to start saving by opening a savings account for them.
If they are younger, you can still make savings "real" to them by having them build their savings in a piggy bank or clear jar. You can motivate them to allot a portion of their allowance to their savings. If you have multiple children, one way to keep them motivated is by giving a prize to whoever earned the highest amount in their savings.

4. Explaining the value of spending money can also be done at home.
You can assign some household chores and pay a small amount once they were able to do it. This will help them realize that money is not earned easily and should be spent wisely.

5. Show your children how an ATM machine works.
While many children know that money doesn't grow on trees, they may think it comes out of a wall. Help your kids understand that you must put money in the bank before you can take it out.

When you discuss money with kids, you help them develop a sense of limits. You're teaching kids that the family has to make choices about how it can spend money. There's only so much money to go around -- if you spend it on some things, you won't have it to spend on others. Teaching your children about saving money doesn't have to be a difficult task. Remember to be patient and consistent, and your children will be able to learn this critical skill in an easy and fun way.