Question

How do you solve `2/3 x +4=3/5x-2`?

Answer 1

`x=-90`

Explanation:

`color(green)([(2x)/3color(red)(xx1)]+[4color(red)(xx1)]=[(3x)/5color(red)(xx1)]-[2color(red)(xx1)] )`

`color(green)([(2x)/3color(red)(xx5/5)]+[4color(red)(xx15/15)]=[(3x)/5color(red)(xx3/3)]-[2color(red)(xx15/15)] )`

`" "(10x)/15" "+" "60/15" "=" "(9x)/15" "-" "30/15`

Multiply all of both sides by 15

`10x+60=9x-30`

`10x-9x=-30-60`

`x=-90`
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Check:

`LHS->2/3x+4`

`" "=" " 2/3(-90)+4" "=" "-56`

`RHS->3/5x-2`

`" "=" "3/5(-90)-2" "=" "-56`

`LHS=RHS" thus true"`

Answer 2

Here are two methods of solution.

Explanation:

Method 1

`2/3 x +4=3/5x-2` is of the form

`A x +4=Bx-2`.

We'll collect terms involving `x` on the left and terms that do not involve `x` on the right.

Subtract `4` and `Bx` (really `3/5x`) from both sides to get

`2/3x-3/5x = -2-4`

Now we have to do the arithmetic of `2/3-3/5`.

Get a common denominator: `10/15 - 3/15`

Simplify `1/15`. This is the coefficient of `x`.

`10/15x-9/15x=-6`

`1/15x = -6`.

Finish by multiplying both sides by `15`

`x = -90`

Method 2

`2/3 x +4=3/5x-2`

Instead of working with the fractional coefficients, we can "clear the fractions".

The least common denominator of the fractions is `15`, so we'll multiply both sides by `15`.

`15(2/3 x +4) = 15(3/5x-2)`

Distribute.

`15/1 2/3 x +15*4 = 15/1*3/5x-15*2`

(I have chosen to write `15/1` in front of the fractions to simplify reducing the fractions.)

`cancel(15)^5 /1 2/cancel(3)_1 x +15*4 = cancel(15)^5/1*3/cancel(5)_1 x-15*2`

`10x + 60 = 9x -30`.

Many of us think that this looks easier to solve.

`10x -9x=- 60 -30`.

`x = -90`.

Another description of Method 2

In `2/3 x +4=3/5x-2`, we have four terms. Let's write them all as fractions with the same denominator.

The common denominator is `15`, so we'll write all four terms as fractions with denominator `15`. (We've gone through most of the arithmetic, so I'll just write the next line.)

`(10x)/15 + 60/15 = (9x)/15 - 30/15`.

Now this is the same as

`(10x + 60)/15 = (9x - 30)/15`.

Since the denominators are the same, these are equal exactly when the numerators are equal:

`10x + 60 = 9x -30`.

`10x -9x=- 60 -30`.

`x = -90`.