Solving Elimination Problems

Solving Elimination Problems-27
The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation. The equations of a system are independent if they do not share ALL solutions.

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If your variable drops out and you have a FALSE statement, that means your answer is no solution.

If your variable drops out and you have a TRUE statement, that means your answer is infinite solutions, which would be the equation of the line.

If the system in two variables has one solution, it is an ordered pair that is a solution to BOTH equations.

In other words, when you plug in the values of the ordered pair it makes BOTH equations TRUE.

But first, we have to learn how to work with systems in general.

That is why we use generic variables likeat this point.Here, the first block is the coefficients of the equations, one equation per row.The second block is the variables, one per row and listed in the same order as the coefficients.A consistent system is a system that has at least one solution.An inconsistent system is a system that has no solution.If you do get one solution for your final answer, is this system consistent or inconsistent?If you said consistent, give yourself a pat on the back!In other words, it is where the two graphs intersect, what they have in common.So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.In this case you can write down either equation as the solution to indicate they are the same line. To remove fractions: since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions. If one of the equations is already solved for one of the variables, that is a quick and easy way to go.If you need to solve for a variable, then try to pick one that has a 1 as a coefficient.

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