Overview: Solving Linear Systems
In order to solve systems of two sentences, the solution has to be true for both sentences. For example, the solution of a problem such as 2x-3y=13 and 3x +y = 3 has to work fox x and y all throughout the problem. Linear systems can be solved by graphing, by substitution, and by addition.
Solving by Graphing
A number sentence that is linear can be graphed along a straight line that contains all the ordered pairs (x, y) that are contained in its solution set. The second sentence in the system (in this example, 3x +y = 3) can also be graphed along its straight line. The point at which both lines intersect is the solution for both sentences.
Solving by Substitution
However, if the student has neither graph paper handy or a graphing calculator to find the exact point where lines intersect, there is another way to solve the problem, by substituting one variable in the first sentence with the definition in the second sentence.
For example, 2x-3y = 13, 3x +y = 3. We already know from the second sentence that y=3 -3x by subtracting 3x from each side of the equation, as in 3x-3x +y = 3 – 3x.
By process of substitution, 2x -3(3 – 3x) = 13 , or 2x – 9 -9x =13, or 2x – 9 + 9x = 13 + 9 (because a negative of a negative equals a positive), or 11x = 22. Therefore, x = 2. Using the second equation, 6 + y = 3, or y = 6 – 3, or y= -3.
Checking, 4 + 9 (as a negative times a negative is a positive) =13.
Solving by Addition
In this example, solve the pair x – 2y = 7, x + y =-2. Adding them together from left to right x + x – 2y + y = 7-2 , or 2x -y = 5. If x equals 1, then y equals -3, because 1 +6 equals 7, making the first pair true. In the second pair, 1-3 equals -2, which makes that also true. Checking the addition sentence, 2+3 (as the double negative changes to a positive) equals 5.
Consistent, Inconsistent, Dependent Systems
The definition of consistent, inconsistent, and dependent systems refers back to the coordinate plane. If two equations have only one point in common, then they are consistent. For example, 2x – 3y = 13 and 3x +y =3, are consistent as they only have one point in common, (2, -3). Similarly, the equations x – 2y =7 and x + y =-2 are consistent, as they have one point in common (1, -3). However, the equations x +y =3 and x +y =-4 have no points in common, and there is no solution set in real space that will solve both equations, because there are no combinations of x and y that will simultaneously solve both equations. They are inconsistent. Dependent systems have an infinite number of points in common, such as x +y =3 and 2x +2y = 6,
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