Question

How do you solve `\frac { 2} { 5} x + \frac { 6} { 5} = \frac { 1} { 4} x - 1`?

Answer 1

`x=-44/3`

Explanation:

`2/5x+6/5=1/4x-1 iff 2/5x-1/4x=-1-6/5 iff`

`8/20x-5/20x=-5/5-6/5 iff 3/20x=-11/5 iff`

`x=-11/5*20/3=-11/1*4/3=-44/3`

so `x=-44/3`

Answer 2

Step 1: Convert all terms to the same denominator
The least common multiple of the denominators as given (that is the LCD of 5 and 4) is 20,
so we will express the terms with denominators of 20:
#{: ("terms in original form: ",,2/5x,6/5,1/4x,1), (,"=",,,,), ("with denominators equal to 20: ",,8/20x,24/20,5/20x,20/20) :}#

Step 2: Rewrite the original equation with the new denominator forms
`8/20x+24/20=5/20x-20/20`

Step 3: Simplify by multiplying everything by the value of the common denominator (20)
`8x+24=5x-20`

Step 4: Subtract 5x from both sides (to isolate x terms on the left)
`3x+24=-20`

Step 5: Subtract 24 from both sides (to isolate the constant terms on the3 right)
`3x=-44`

Step 6: Divide both sides by 3 (to reduce the equation to a single x)
`x=-44/3`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Verification: If `x=-44/3`

L.S.`=2/5(-44/3)+6/5`

`color(white)("XX")=(-88/15)+18/15`

`color(white)("XX")=-70/15`

`color(white)("XX")=-14/3`

R.S.`=1/4(-44/3)-1`

`color(white)("XX")=(-11/3)-3/3`

`color(white)("XX")=-14/3`

L.S. = R.S.
so our result is correct.