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the product of twice a number and two is the same as the difference of three times the number and 7 over 9. find the number

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2 Answers

Try to break off a word or two at a time.

the product of twice a number and two is the same as the difference of three times the number and 7 over 9. find the number

Product means multiplication, so you need a multiplication sign *. Now find what you're multiplying. There are two things being multiplied together:

the product of twice a number and two is the same as the difference of three times the number and 7 over 9. find the number

Whenever you see "a number", make up a variable. Let's use x. When you see "twice" that means 2 *. so now we have 2 * x.

The other number being multiplied is 2.

Let's put all that together.

the product of twice a number and two is the same as the difference of three times the number and 7 over 9. find the number

We can rewrite all of those words as ( 2 * x ) * 2.

Now let's look at the next chunks.

( 2 * ) * 2 is the same as the difference of three times the number and 7 over 9. find the number

When you see "is the same as", that usually means an equals sign =. Now we look at what comes after the equals sign.

( 2 * x ) * 2the difference of three times the number and 7 over 9. find the number

"The difference" means a minus sign -. We need two things for subtraction:

( 2 * x ) * 2 = the difference of three times the number and 7 over 9. find the number

"Three times" is 3 *, and "the number" is the same variable x that we picked before. So "three times the number" is the same as 3 * x.

"7 over 9" is a fraction, 7/9.

Let's put that part together.

( 2 * x ) * 2 = the difference of three times the number and 7 over 9. find the number

We can rewrite those words as ( 3 * x ) - ( 7/9 ).

Now our problem looks like this:

( 2 * x ) * 2 = ( 3 * x) - ( 7/9 ). find the number

And that's algebra.

( 2 * x ) * 2 = ( 3 * x ) - ( 7/9 )

Since the left hand side is all multiplication, you can move around the factors and parentheses, which is good, since you know what ( 2 * 2 ) is and can simplify that part of the equation.

( 2 * 2 ) * x = ( 3 * x ) - ( 7/9 )

4 * x = (3 * x) - ( 7/9 )

When we're doing algebra with variables like x, we can just decide not to write the multiplication signs. Then our equation would look like:

4x = 3x - 7/9

Put the mulitples of x on one side of the equation. One way to do that would be:

4x - 3x = - 7/9

Then combine like terms:

x = - 7/9

So we found the number! It's -7/9!

Let x represent the number you are looking for.

Then the first part, which states that the product of twice a number (x) and two, would end up looking like the following:

                       (2*x) * (2) = (2*2) * x = 4x

For the second part, which states that the difference of three times the number (x) and 7 over 9 (7/9), you would end up at the following:

                                             (3*x) - (7/9) = 3x - (7/9)

Since it says that the first part (4x) is the same as the second part (3x - (7/9)), then you would set these two equations equal to one another and solve for x.

That is,     4x = 3x - (7/9)          

Combine like terms and separate them to one side of the equation; in this case, subtract 3x from both sides of the equation....

         4x - 3x  =  3x - 3x - (7/9)   ===>   1x = 0 - (7/9)   ===>   x = -7/9