Let's solve the problem step-by-step where we need to find the value of [tex]\( m \)[/tex] such that [tex]\( x = 9 \)[/tex] is a root of the equation [tex]\((6m - 1)x^2 - (2m + 5)x + 3 = 0\)[/tex].
### Step-by-Step Solution
1. Substitute [tex]\( x = 9 \)[/tex] into the equation:
[tex]\[
(6m - 1)(9^2) - (2m + 5)(9) + 3 = 0
\][/tex]
2. Calculate [tex]\( 9^2 \)[/tex]:
[tex]\[
9^2 = 81
\][/tex]
3. Insert [tex]\( 81 \)[/tex] for [tex]\( 9^2 \)[/tex]:
[tex]\[
(6m - 1) \cdot 81 - (2m + 5) \cdot 9 + 3 = 0
\][/tex]
4. Distribute the terms in the equation:
[tex]\[
81 \cdot (6m - 1) - 9 \cdot (2m + 5) + 3 = 0
\][/tex]
[tex]\[
81 \cdot 6m - 81 - 18m - 45 + 3 = 0
\][/tex]
5. Simplify the equation:
[tex]\[
486m - 81 - 18m - 45 + 3 = 0
\][/tex]
6. Combine like terms:
[tex]\[
486m - 18m - 81 - 45 + 3 = 0
\][/tex]
[tex]\[
468m - 123 = 0
\][/tex]
7. Isolate [tex]\( m \)[/tex]:
[tex]\[
468m = 123
\][/tex]
8. Solve for [tex]\( m \)[/tex]:
[tex]\[
m = \frac{123}{468}
\][/tex]
9. Simplify the fraction:
[tex]\[
m = \frac{41}{156}
\][/tex]
Therefore, the value of [tex]\( m \)[/tex] that makes [tex]\( x = 9 \)[/tex] a root of the equation [tex]\((6m - 1)x^2 - (2m + 5)x + 3 = 0\)[/tex] is:
[tex]\[
m = \frac{41}{156}
\][/tex]
