Question

Solve the equation (linear equation)


`4^(x-7)` ×
`8^(2x-3) =(32)/(2^(x-9) )`

Answer 1

`x=4(1)/(9)`

Explanation:

We are given the following linear equation and we are to solve it:

`4 ^ { x - 7 } * 8 ^ { 2x - 3 } = \frac { 32 } { x ^ { x - 8 } }`

Changing the constants to the same base to make it easier to solve:

`2 ^ { 2 ( x - 7 ) } * 2^ { 3 ( 2x - 3 ) } = \frac { 2 ^ 5 } { 2 ^ { x - 9 } }`

`2^(2x-14) * 2^(6x-9) = 2^(5)(2^(-x+9))`

`2 ^ { 2x - 14 + 6x - 9 } = 2 ^ { 5 - x + 9 }`

`2 x + 6x - 14 - 9 = 5 - x + 9`

`8x+x=14+9+5+9`

`9x=37`

`x=4(1)/(9)`

Answer 2

Answer:
`x=(37)/(9)`

Explanation:

By the negative exponent rule, you have that:

`((1)/(a))^n=a^(-n)`

By the exponents properties, you know that:

`(m^n)^l=m^((nl))`

`(m^n)(m^l)=m^((n+l))`

Rewrite 4, 8 and 32 as following:

4=2²

8=2³

32=2⁵

Rewrite the expression:

`(2^2)^((x-7))*(2^3)^((2x-3))=(32)/(2^((x-9)))`

Keeping on mind the exponents properties, you have:

`(2)^(2(x-7))*(2)^(3(2x-3))=32(2^(-(x-9))`

`(2)^(2(x-7))*(2)^(3(2x-3))=(2^5)(2^(-(x-9)))\\\\(2)^((2x-14))*(2)^((6x-9))=(2^5)(2^((-x+9)))\\\\2^(((2x-14)+(6x-9)))=2^((5+(-x+9)))`

As the bases are equal, then:

`(2x-14)+(6x-9)=5+(-x+9)\\\\2x-14+6x-9=5-x+9\\\\8x-23=14-x\\9x=37`

`x=(37)/(9)`