How to Do Linear Substitution
Linear substitution is a method of solving systems of linear equations. It is used to find the intersection of linear equations, or lines. Linear equations may be in slope intercept form, y = mx + b, or in standard form, Ax + By = C. In linear substitution, you combine the two equations through algebraic manipulation to find the point that the lines both share. Not all systems of linear equations have solutions because not all lines intersect. Parallel lines are an example of lines that do not intersect.
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Instructions
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1
Isolate a variable in one of the equations in the system of linear equations. For example, given the equations 8x + 2y = 6 and 6x + 8y = 4, pick 8x + 2y = 6 and solve for "y." Subtract the 8x term from both sides to get 2y = 6 -- 8x. Then divide by 2 to get y = 3 -- 4x.
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2
Substitute this in for the corresponding variable in the other equation. In the example, substitute 3 -- 4x for y in 6x + 8y = 4. So, 6x + 8(3 -- 4x) = 4.
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3
Solve for the other variable. In the example, distribute the 8 so you have 6x + 24 - 32x = 4. Then combine the like terms so that you have 24 - 26x = 4. Subtract 24 from both sides to get -26x = -20 and then divide by -26 to get x = 10/13.
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4
Plug this solution back into the first equation and solve for the other variable. In the example, plug x = 10/13 into 8x + 2y = 6 to get 8(10/13) + 2y = 6. Simplifying gives y = -1/13.
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5
Use the results from Step 3 and Step 4 and write the answer, which is the intersection of the lines, as a point in the form (x, y). In the example, the point is (10/13, -1/13).
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