Certainly! Let's solve the equation step-by-step to find the value of [tex]\( u \)[/tex].
Given the equation:
[tex]\[ 18u - 51 = 9(4u + 5) - 6(3u - 10) \][/tex]
1. Distribute the constants on the right-hand side:
First, distribute [tex]\( 9 \)[/tex] in [tex]\( 9(4u + 5) \)[/tex]:
[tex]\[ 9(4u + 5) = 9 \cdot 4u + 9 \cdot 5 = 36u + 45 \][/tex]
Next, distribute [tex]\( 6 \)[/tex] in [tex]\( 6(3u - 10) \)[/tex]:
[tex]\[ -6(3u - 10) = -6 \cdot 3u + (-6) \cdot (-10) = -18u + 60 \][/tex]
So, substituting these into the original equation gives:
[tex]\[ 18u - 51 = 36u + 45 - 18u + 60 \][/tex]
2. Combine like terms on the right-hand side:
Combine the [tex]\( u \)[/tex]-terms and the constant terms separately on the right-hand side:
[tex]\[ 36u - 18u + 45 + 60 = 18u + 105 \][/tex]
So the equation simplifies to:
[tex]\[ 18u - 51 = 18u + 105 \][/tex]
3. Isolate the variable [tex]\( u \)[/tex]:
To isolate [tex]\( u \)[/tex], let's first cancel out the [tex]\( 18u \)[/tex] terms on both sides by subtracting [tex]\( 18u \)[/tex] from each side:
[tex]\[ 18u - 18u - 51 = 18u - 18u + 105 \][/tex]
[tex]\[ -51 = 105 \][/tex]
This results in a statement [tex]\( -51 = 105 \)[/tex], which is a contradiction.
Since the equation leads to a contradiction, it implies there are no values of [tex]\( u \)[/tex] that satisfy the given equation. Therefore, the equation has no solution.
So, the solution is:
[tex]\[ \boxed{\text{No solution}} \][/tex]
