When dealing with a system of linear equations there are two methods to algebraically solve the question. One is substitution and the other is elimination which is meant to be a shortcut. Both methods will bring you to the same solution but with more practice, you will recognize patterns and see which method would work best when given a system. The best way to show how to solve these kinds of questions are by providing an example to work on.
Example: Line 1: 2x + y = 6
Line 2: 6x + 2y = 4
Substitution Method:
This method involves isolating for one variable (x/y) of Line 1 then substituting that variable into Line 2. This will allow you to isolate and solve for the other variable (y/x). Once you have the x- and y- coordinates, you then have the solution which is the point of intersection between the two lines.
- Line 1: y = 6 – 2x
2. Substitute into Line 2: 6x + 2(6 – 2x) = 4
6x + 12 – 4x = 4
2x = -8
x = -4
3. Substitute back into Line 1 or 2: y = 6 – 2(-4)
y = 14
4. Solution: Point of intersection is (-4, 14)
Elimination Method:
This method involves “eliminating” one variable by finding the lowest common multiple for a chosen variable. Then you would put both lines as if you are adding or subtracting to find a final line just as you would for adding or subtracting large numbers. This removes leaves one variable to solve. This variable can then be substituted back into the other line to find the remaining variable.
- Line 1: 2x + y = 6 (x2) *The ‘y’ variable seems like the simplest to
Line 2: 6x + 2y = 4 (x1) get the lowest common multiple of.
2. Line 1: 4x + 2y = 12
Line 2: 6x + 2y = 4
Final: -2x + 0y = 8
*Now that the ‘y’ variable are the same. We either have to subtract or add the variable to get 0y. In this case, we subtract.
3. Solve: -2x = 8
x = -4
4. Substitute back into Line 1 or 2: y = 6 – 2(-4)
y = 14
5. Solution: Point of intersection is (-4, 14)
There you go, both methods get you the same answer whenever asked to solve a linear system. Notice that when x or y has no coefficient, then substitution would be faster. If the x or y of both lines are the same then elimination would be faster.
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