Overview:
Many equations use more than one variable, because there is more than one unknown in the sentence. Many times, one variable is expressed in terms of the other variable. Solving for one variable clears the way to solve for the other variable.
Why Use Two Different Variables?
Sometimes there are two different conditions that have to be fulfilled, and each variable represents a different condition. For example, suppose that one condition is that there are 200 more crates of tomatoes than green peppers in a warehouse. Let x equal the number of crates of tomatoes, and y equal the number of crates of green peppers. Then the first condition would be expressed by the equation x-y =200. The second condition is that twice the number of crates of tomatoes is 100 more than three times the number of crates of green peppers. That would be expressed by the equation 2x -100 = 3y.
How Are the Equations Related?
At this point, it helps to write the equations close together, so that x -y = 200 and 2x – 100 =3y. With the equations closer together, it is more easy to see the relationships between them. The equation x-y= 200 can be changed by adding y to each side so that x-y + y =200 +y. The -y and y cancel each other out, so that x = 200 +y.
Substituting Expressions
Now that there is a value for x in terms of y as 200 + y, it can be substituted in the second equation, so that 2(200 +y) -100 =3y, or 400 + 2y – 100 =3y. If 300 +2y = 3y, then 300 +2y- 2y = 3y-2y, or 300 = y. Solving for x, if x = 200 +y, then x = 200 + 300 or x = 500.
Do the Equations Check?
In Condition 1, 500 – 300 = 200. In Condition 2, does 2(500) – 100 = 3(300)? Condition 2 checks because 1000 – 100 = 900, so there are 500 crates of tomatoes and 300 crates of peppers in the warehouse.
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