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How Do You Solve for a Variable in a Fraction Using Multiplication First?

Solve the equation (9t + 27)/2 = 45 for t.

Summary

  1. 't' is the variable we are trying to solve for
  2. Dividing by 2 is the same as multiplying by 1/2
  3. We can multiply by the reciprocal of 1/2, 2, to get rid of the division on the left hand side

Notes

    1. (9t + 27)/2 is the same as (1/2)(9t + 27)
    2. 't' is the variable we are trying to solve for
    1. The Multiplication Property of Equality:
    2. If you multiply one side by a number, you MUST multiply the other side by the number
    3. Otherwise, the equation does not stay equal
    1. To find a number's reciprocal, just switch the numerator and denominator
    2. So 1/2 would become 2/1, which reduces to 2
    3. Reciprocals cancel each other out when you multiply them
    4. So 2 and 1/2 will cancel out in this case
    1. The Subtraction Property of Equality:
    2. If you subtract a number from a side, you MUST also subtract that number from the other side
    3. Otherwise, the equation does not stay equal
    1. The 27's cancel out on the left, leaving us with 9t
    2. On the right, 90-27=63
    1. The Division Property of Equality:
    2. If you divide one side by a number, you MUST divide the other side by the same number
    3. Otherwise, the equation does not stay equal
    1. The 9's cancel out on the left, leaving us with just 't'
    2. Dividing 63 by 9 gives us 7, which is our value for 't'!
    1. Plug in t for 7 into the original equation to check that it's the right answer
    2. If plugging in 7 gives us a true statement, we know t=7 is the right answer
    1. If we get a true statement, we know the answer is correct
    2. The original equation was (9t + 27)/2 = 45
    1. According to the order of operations, first we need to simplify what's in the numerator
    2. So we need to multiply 9•7 first to get 63
    3. Then we can add 63 and 27 to get 90
    4. Once the numerator is simplified down to 90, we can divide by 2 to get 45
    5. 45 was what we had on the right hand side in the original equation, so we get a true statement!
    6. Since plugging in t=7 gave us a true statement, we know that 't' does equal 7