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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