If [tex]$x=\frac{a+3}{2 a}-1$[/tex], express [tex]$\frac{2 x+1}{3 x+1}$[/tex] in terms of a
Answer (1)
First, simplify x in terms of a : x = 2 a 3 − a .
Substitute the expression for x into 3 x + 1 2 x + 1 .
Simplify the resulting expression: 3 ( 2 a 3 − a ) + 1 2 ( 2 a 3 − a ) + 1 = 2 a 9 − 3 a + 1 a 3 − a + 1 .
The final expression is 9 − a 6 .
Explanation
Understanding the Problem We are given the equation x = 2 a a + 3 − 1 and we want to express 3 x + 1 2 x + 1 in terms of a .
Simplifying x First, simplify the expression for x in terms of a :
x = 2 a a + 3 − 1 = 2 a a + 3 − 2 a = 2 a 3 − a .
Substituting x Next, substitute the simplified expression for x into 3 x + 1 2 x + 1 :
3 x + 1 2 x + 1 = 3 ( 2 a 3 − a ) + 1 2 ( 2 a 3 − a ) + 1 .
Simplifying the Expression Now, simplify the resulting expression: 3 ( 2 a 3 − a ) + 1 2 ( 2 a 3 − a ) + 1 = 2 a 9 − 3 a + 1 a 3 − a + 1 = 2 a 9 − 3 a + 2 a a 3 − a + a = 2 a 9 − a a 3 = a 3 ⋅ 9 − a 2 a = 9 − a 6 .
Final Answer Therefore, the expression 3 x + 1 2 x + 1 in terms of a is 9 − a 6 .
Examples
This problem demonstrates how to substitute and simplify algebraic expressions, a skill used in various fields. For example, in physics, you might have a formula for velocity in terms of time and need to express kinetic energy in terms of velocity. By substituting and simplifying, you can find a direct relationship between kinetic energy and time.
