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Given:
[tex]\[
\left(\frac{a}{b}\right)^{x-1}=\left(\frac{b}{a}\right)^{x-3}
\][/tex]

Solve for [tex]\( x \)[/tex].

Sagot :

GinnyAnsweridn
To solve the equation [tex]\(\left(\frac{a}{b}\right)^{x-1} = \left(\frac{b}{a}\right)^{x-3}\)[/tex] for [tex]\(x\)[/tex], we need to follow these steps:

1. Rewrite the right-hand side: Observe that we can express [tex]\(\left(\frac{b}{a}\right)^{x-3}\)[/tex] using the reciprocal:
[tex]\[ \left(\frac{b}{a}\right)^{x-3} = \left(\frac{a}{b}\right)^{-(x-3)} \][/tex]

2. Set the exponents equal: Now that both sides of the equation are written as powers of [tex]\(\frac{a}{b}\)[/tex], we can set the exponents equal to each other since the bases are the same:
[tex]\[ (x-1) = -(x-3) \][/tex]

3. Solve the resulting equation:
[tex]\[ x - 1 = -x + 3 \][/tex]
Add [tex]\(x\)[/tex] to both sides to gather all [tex]\(x\)[/tex] terms on one side:
[tex]\[ x + x - 1 = 3 \][/tex]
[tex]\[ 2x - 1 = 3 \][/tex]
Add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 2x = 4 \][/tex]
Divide by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 2 \][/tex]

Thus, the solution to the equation is:
[tex]\[ \boxed{2} \][/tex]
Wegnerkolmp2741oidn

Answer:

x=2

Step-by-step explanation:

(a/b) ^ (x-1) = ( b/a) ^ (x-3)

We can rewrite the right hand side because m^ n = 1/m ^-n

( b/a) ^ (x-3) = ( 1/ ( b/a)) ^ -( x-3) = (a/b) ^ ( 3-x)

(a/b) ^ (x-1) = ( a/b)) ^ (3-x)

Since the bases are now the same, the exponents must be the same.

x-1 = 3-x

Add x to each side.

x-1+x = 3-x+x

2x -1 = 3

Add 1 to each side.

2x-1+1 =3+1

2x=4

Divide by 2.

2x/2 = 4/2

x=2

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