Question

b. solve for p: 8p + 3w = w - 15 - 4p
set 42

1. war eagle spied a total of 1428 antelope and wildebeests grazing on the savannah. if the ratio of antelope to wildebeests was 9 to 5, how many antelope were there?

Explanation:

Step1: Solve for \( p \) in \( 8p + 3w = w - 15 - 4p \)

First, we want to get all the terms with \( p \) on one side and the other terms on the opposite side. Add \( 4p \) to both sides of the equation:
\( 8p + 4p+ 3w = w - 15 - 4p+ 4p \)
Simplify the left side (combining like terms for \( p \)) and the right side (the \( -4p \) and \( +4p \) cancel out):
\( 12p + 3w = w - 15 \)
Next, subtract \( 3w \) from both sides to isolate the terms with \( p \) on the left:
\( 12p + 3w - 3w= w - 3w - 15 \)
Simplify both sides:
\( 12p = -2w - 15 \)
Now, divide both sides by 12 to solve for \( p \):
\( p=\frac{-2w - 15}{12}=\frac{-w}{6}-\frac{5}{4} \)

Step2: Solve the antelope - wildebeest problem

Let the number of antelope be \( 9x \) and the number of wildebeests be \( 5x \) (since the ratio of antelope to wildebeests is \( 9:5 \)). The total number of antelope and wildebeests is 1428. So we can set up the equation:
\( 9x + 5x=1428 \)
Combine like terms on the left side:
\( 14x = 1428 \)
Divide both sides by 14 to solve for \( x \):
\( x=\frac{1428}{14} = 102 \)
Now, the number of antelope is \( 9x \), so substitute \( x = 102 \) into \( 9x \):
Number of antelope \(=9\times102 = 918\)

Answer:

For the equation \( 8p + 3w = w - 15 - 4p \), \( p=\boldsymbol{\frac{-w}{6}-\frac{5}{4}} \) (or simplified as \( p =-\frac{w + \frac{15}{2}}{6}\) but the first form is more standard).
For the antelope - wildebeest problem, the number of antelope is \(\boldsymbol{918}\).