Question

How do you solve and check `\frac { x + 8} { 16} = \frac { 9} { 8} + \frac { x - 6} { 7}`?

Answer 1

`color(blue)(ul(bar(abs(color(black)(x=26/9))))`

Explanation:

We have some arithmetic with fractions. When doing addition and subtraction, we need to make sure the denominators are all the same (it's like we're doing math with slices of pizza and it's the denominator that tells us the size of the slice).

To get the denominator all the same number, let's find the Lowest Common Multiple:

`16=2xx2xx2xx2`
`8=2xx2xx2`
`7=7`

So our common multiple will have four 2s and a 7:

`2xx2xx2xx2xx7=112`

So we can say:

`(x+8)/16=9/8+(x-6)/7`

`(x+8)/16(7/7)=9/8(14/14)+(x-6)/7(16/16)`

`(7(x+8))/112=126/112+(16(x-6))/112`

`7x+56=126+16x-96`

`7x+56color(red)(-7x-126+96)=126+16x-96color(red)(-7x-126+96)`

`56-126+96=16x-7x`

`26=9x`

`color(blue)(ul(bar(abs(color(black)(x=26/9))))`

And now let's check:

`(x+8)/16=9/8+(x-6)/7`

`(26/9+8)/16=9/8+(26/9-6)/7`

`(26/9+72/9)/16=9/8+(26/9-54/9)/7`

`(98/9)/16=9/8+(-28/9)/7`

`(98/9)/16(7/7)=9/8(14/14)+(-28/9)/7(16/16)`

`((686/9)/112)=126/112+(-448/9)/112`

`686/9=126(9/9)-448/9`

`686/9=1134/9-448/9`

`686/9=686/9 color(white)(000)color(green)sqrt`