Overview
One way to solve systems of equations with more than one variable is to substitute values of one variable for another. Suppose that a variable y is equal to 2x. One way to solve the equation x + y = 12 is by substituting x + 2x = 12 so that x is equal to 4 and y is equal to 2∙4, or 8.
Substituting Variables
The simplest form of substitution is when one variable is listed in terms of the other, as in the example above. Suppose that y = x + 3 and x + y = 13. Then x + x + 3 = 13, so 2x equals 10, or x equals 5. If x equals 5, then x +3 equals 8 and 5 + 8 = 13. Checking the equation by replacing the variables with numbers fulfills both conditions.
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Figure 1: Substituting values for variables is like fitting together the pieces of a puzzle. |
Substituting Quantities
Another form of substitution occurs when two different variables are given in terms of a third, as when a mixture or solution has one quantity of one variable and another quantity of another variable and the parts added together make a given amount. Suppose that a fruit punch contains 3 parts apple juice to 1 part cranberry juice. If 5 gallons of punch are made, how many quarts of apple juice and how many quarts of cranberry juice will it take? There are 2 substitutions that take place in this problem. If there are 4 quarts in a gallon, then there are 20 quarts in 5 gallons. Let a stand for apple juice and c for cranberry juice. Then a = 3c, and 3c + c = 20. If 4c = 20 then c = 5, and 3c = 15.
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Figure 2: Combining quantities is another way to use substitution. |
Substituting Equations
Another form of substitution occurs when two equations are equivalent to the same variable. Suppose that there are 2 lines, one with y = 3x +2 and one with y = -4x – 5. Will they intersect at any point in the coordinate plane? One of the ways to solve this problem is to set 3x + 2 = -4x – 5. With 3x + 4x = -2 + -5, 7x = -7, so x = -1. Choosing the first equation, the y coordinate is -1.
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Figure 3: Use substitution to find out where lines intersect. |
Equations with No Solution
Suppose the lines in question were y = 3x + 3 and y = 3x + 2. Will those lines intersect at any point? As before, 3x + 3 = 3x + 2. In solving the equation the x cancels out, so that the resulting equation gives a nonsense statement 0 = -1. The nonsense statement means that there is no point that the lines intersect. The lines are parallel.
Figure 4: If there is no solution, graphed lines are parallel.
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