Suppose $8b - 6b - 2q = 16$, $0 = q + 5$. Let $k$ be $(12 - 7 - 8) + 3$. Suppose $h - 32 = -h + 2g$, $-5g - 20 = 0$. Solve $bs + ks = h$ for $s$.

Understand the Problem

The problem presents a series of equations that need to be solved. First we need to solve for the variables q, k, g, and h with the given individual equations. Once we have the values for k and h, we can substitute them into the final equation bs + ks = h and solve for s in terms of b.

Answer

$s = 4$

Steps to Solve

Solve for $q$ We are given the equation $0 = q + 5$. Subtract 5 from both sides to solve for $q$: $q = -5$
Solve for $b$ Substitute the value of $q$ into the equation $8b - 6b - 2q = 16$: $2b - 2(-5) = 16$ $2b + 10 = 16$ $2b = 16 - 10$ $2b = 6$ $b = 3$
Solve for $k$ Calculate the value of $k$ using the given expression: $k = (12 - 7 - 8) + 3$ $k = (5 - 8) + 3$ $k = -3 + 3$ $k = 0$
Solve for $g$ Solve the equation $-5g - 20 = 0$ for $g$: $-5g = 20$ $g = \frac{20}{-5}$ $g = -4$
Solve for $h$ Substitute the value of $g$ into the equation $h - 32 = -h + 2g$: $h - 32 = -h + 2(-4)$ $h - 32 = -h - 8$ $2h = 32 - 8$ $2h = 24$ $h = 12$
Solve for $s$ Substitute the values of $k$ and $h$ into the equation $bs + ks = h$: $bs + (0)s = 12$ $bs = 12$ $s = \frac{12}{b}$
Substitute b = 3 to find the final answer $s = \frac{12}{3}$ $s = 4$

$s = 4$

More Information

The problem required solving a series of equations to find the values of intermediate variables, and then substituting those values into a final equation to solve for the desired variable $s$.

Tips

A common mistake is making arithmetic errors when simplifying the expressions for $k$, $g$, and $h$. Another mistake is incorrectly substituting the values into the final equation when solving for $s$. It's important to take care performing each step, and double check your work, especially with basic arithmetic.

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