To solve the given equation [tex]\( 1 - 4p + 6p = 5(-3 + 4p) - 6(1 + 3p) \)[/tex], let’s go through the steps systematically:
1. Combine like terms on the left-hand side:
[tex]\[
1 - 4p + 6p = 1 + 2p
\][/tex]
So the equation now looks like this:
[tex]\[
1 + 2p = 5(-3 + 4p) - 6(1 + 3p)
\][/tex]
2. Distribute the constants on the right-hand side:
[tex]\[
5(-3 + 4p) = 5 \cdot (-3) + 5 \cdot 4p = -15 + 20p
\][/tex]
[tex]\[
-6(1 + 3p) = -6 \cdot 1 - 6 \cdot 3p = -6 - 18p
\][/tex]
Therefore, the equation now becomes:
[tex]\[
1 + 2p = -15 + 20p - 6 - 18p
\][/tex]
3. Combine like terms again on the right-hand side:
[tex]\[
-15 - 6 + 20p - 18p = -21 + 2p
\][/tex]
So, we now have:
[tex]\[
1 + 2p = -21 + 2p
\][/tex]
4. Subtract [tex]\(2p\)[/tex] from both sides:
[tex]\[
1 + 2p - 2p = -21 + 2p - 2p
\][/tex]
Simplifying, we get:
[tex]\[
1 = -21
\][/tex]
This statement [tex]\(1 = -21\)[/tex] is obviously a contradiction and is never true. Thus, there is no value of [tex]\( p \)[/tex] that can satisfy the given equation.
Conclusion:
There are no solutions to the equation [tex]\( 1 - 4p + 6p = 5(-3 + 4p) - 6(1 + 3p) \)[/tex].
In other words, the solution set for [tex]\( p \)[/tex] is the empty set: [tex]\( \boxed{[]} \)[/tex].
