Variations on Themes
While it is best to conform to the basic addition and multiplication routines for solving systems of equations, in a few instances, you benefit if you perform a few preliminary activities to make your work easier. The actions you take involve practical measures that you often take when working with fractions.
Change the Order
One measure involves examining the system of equations to discover whether it might be best to switch the order of the equations to make addition and multiplication activities easier. Here is an example involving the system of equations that resembles the one you explored in the previous section:
-4x + 2y = 0
x - 4y = -1
You can make it easier to work with this system of equations if you leave the coefficient of x in the second equation unchanged and instead manipulate the first equation. To make it possible to preserve the coefficient of x in the second equation, you reverse the order of the equations. The second equation becomes the first:
x - 4y = -1
-4x + 2y = 0
The changed order does not alter the value the equations generate. It only makes it so you can work with them more readily. Given this reordering, then, you can proceed with the elimination of the x variable.
Preliminary Multiplications
In some instances, you work with systems of equations that contain decimal values. In such situations, if you inspect the decimal values, you might find that if you multiply them by a power of 10 (10, 100, and so on), you can eliminate the decimal values. Elimination of the decimal values makes it much easier to proceed as you work with multiplications you require to eliminate the x or y variables. Here is an example of a system of equations that contains decimal values:
-0.4x + 0.6y = 0.04
0.02x - 0.4y = 1.4
For both of these equations, if you multiply by 100, you can eliminate the decimal points and so arrive at terms that consist of integers. Your actions take the following form:
| -0.4x + 0.6y = 0.04 | (multiply by 100) |
| 0.02x - 0.4y = 1.4 | (multiply by 100) |
This multiplication results in a new version of the system that preserves the value relationship of the first:
-40x + 60y = 4
2x - 40y = 140
Given this adjusted view of the system of equations, you can now proceed much more readily toward a solution for the system.
Exercise Set 11.3Solve each system of equations. Check your answers.
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