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Solving Equations with Fractions

Q:  How do you solve: 5/7 (x – 70) + 3/7 = 0?

A:  There are many different ways to solve this problem, and all will lead you to the same answer.  I am going to show two methods:

Method 1

Solve as you normally would:

5/7 (x – 70) + 3/7= 0

Subtract 3/7 from both sides:

5/7 (x – 70) + 3/7 -3/7 = 0 -3/7

5/7 (x – 70) = -3/7

Multiply the 5/7 through to the (x – 70) like so:

5/7*x – 5/7 * 70 = -3/7

5/7x – 350/7 = -3/7

Even though -350/7 can be reduced, I am going to wait to do that…

So,

5/7x – 350/7 = -3/7

Now add 350/7 to both sides:

5/7x – 350/7 + 350/7 = -3/7 + 350/7

5/7x = 347/7

Now, the final step to get x by itself is to multiply by the reciprocal of 5/7, which is 7/5… Doing this will clear the fraction:

(7/5) * 5/7x = 347/7 *(7/5)

35/35x = 347*7/(7*5)

x = 347*7/(7*5) [cancel a 7 out of the top and out of the bottom]

x = 347/5

Method 2

Clear the fraction first.  Since “7” is the only denominator, multiply everything by 7:

5/7 (x – 70) + 3/7= 0

7 * [ 5/7 (x – 70) + 3/7= 0 ]

Distribute the “7” through to each term like so:

7*(5/7)(x – 70) + 7(3/7) = 7 *0

Simplify the fractions… Notice we can cancel out most of the 7’s:

7*(5/7)(x – 70) + 7(3/7) = 7 *0

5(x – 70) + 3 = 0

Now, no more fractions!  Distribute the 5 through:

5x – 350 + 3 = 0

Simplify:

5x – 347 = 0

5x = 347

x = 347/5

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