Suppose -3p = -p + m + 60, 3m + 151 = -5p. Let u(d) = 2d² + 58d + 9. Let s be u(p). Solve 3j = -s - 3 for j.
Understand the Problem
The question presents multiple equations that need to be solved step by step. Specifically, it involves solving for variable j given the equations provided for p, m, and s. It requires understanding how to manipulate and solve algebraic expressions.
Answer
The value of \( j \) is \( -4 \).
Answer for screen readers
The value of ( j ) is ( -4 ).
Steps to Solve
- Solve for m in terms of p
Start with the first equation:
$$ -3p = -p + m + 60 $$
Rearranging gives:
$$ m = -3p + p - 60 $$
Then simplify:
$$ m = -2p - 60 $$
- Substitute m into the second equation
Now substitute $m$ into the second equation:
$$ 3m + 151 = -5p $$
Replacing $m$ gives:
$$ 3(-2p - 60) + 151 = -5p $$
Distribute the 3:
$$ -6p - 180 + 151 = -5p $$
- Combine like terms and solve for p
Now combine like terms:
$$ -6p - 29 = -5p $$
Add $6p$ to both sides:
$$ -29 = p $$
- Calculate s using p
Now substitute $p$ into $u(p)$ to find $s$. First, recall:
$$ u(d) = 2d^2 + 58d + 9 $$
Substituting $p$:
$$ s = u(-29) = 2(-29)^2 + 58(-29) + 9 $$
Calculating each term:
- First term: $2(-29)^2 = 2 \cdot 841 = 1682$
- Second term: $58 \cdot (-29) = -1682$
- Third term: $9$
Now sum these up:
$$ s = 1682 - 1682 + 9 = 9 $$
- Substitute s into the equation for j
Now substitute $s$ into the equation $3j = -s - 3$:
$$ 3j = -9 - 3 $$
Thus:
$$ 3j = -12 $$
- Solve for j
Now divide by 3:
$$ j = -4 $$
The value of ( j ) is ( -4 ).
More Information
In this problem, we derived values for variables step by step, utilizing algebraic equations and functions. The computed value of ( j ) shows how algebra can simplify complex relationships between variables.
Tips
One common mistake is miscalculating the terms during substitution, especially with signs and operations. It's essential to double-check each step to avoid propagation of errors.
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