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In solving mathematics problems, one of the powerful approaches is to reduce an unknown problem into a problem that we already know how to solve. Up to this point, we have learned how to solve a linear equation with one variable. In other words, if we can transform the equations with two variables into an equation with just one variable, then we can solve the problem. How can that be done?
Let´s use the following specific problem to show how:
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y + x -3 = 0
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(1)
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3y - 2x = 0
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(2)
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When working with a system of linear equations with two variables, we have two equations. Using the first equation, we can solve for one variable, such as y, in terms of the other variable and real (and honest) number(s), as follows:
You might wonder why we did this, since we still don´t know what y should be. Be patient. The trick is to substitute the variable y in the equation (2) with (3). Thus, the name Substitution Method. The result is an equation with only one variable x:
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3y - 2x = 0
3(-x+3) - 2x = 0
-5x + 9 = 0
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Now, we obtained an equation with only one variable, and the knowledge of how to solve a linear equation can be applied to find the solution.
Therefore,
Once we obtain the value for x, we can substitute (thus the name Substitution Method) the value of x in either original equation or even equation (3) to obtain an honest number for y:
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