Question: Chapter 9G: Using Formulas KEY IDEAS • The subject of an equation is a pronumeral (or variable) that occurs by itself • A formula or rule is an equation containing two or more variables. one of which is the subject of the equation, the formula can be rearranged so that any variable is the subject. The subject of the formula is the variable that sits on its own. y = 2x - 5 To find an unknown, we can substitute numbers for all other variables. • To use a formula, substitute all known values and then solve the equation to find the unknown Example 15 Consider the rule k = 3b + 2. Find the value of: (a) k if b = 5 (b) k if b = 10 (c) k if b = 23 Example 16 Consider the rule Q = w(4 + t). Find the value of: (a) Q if w = 10, t = 3 (b) t if Q = 42, w = 6 (c) Q if w = 2, t = -5 (d) t if Q = 45, w = 3 Example 17 You get paid $25 per hour, and there's a $50 bonus for finishing a shift. The total pay formula is: T = 25h + 50 If you earned $225, how many hours (h) did you work? Example 18 A taxi charges a $5 flat rate plus $2 per kilometre travelled. The formula is: C = 2k + 5 If the total cost C = $21, how many kilometres (k) did you travel?

1. Problem: Use the given formulas to find unknown values by substitution and solving. --- Example 15: Given $$k = 3b + 2$$ (a) Find $$k$$ if $$b = 5$$. Substitute $$b = 5$$: $$k = 3(5) + 2 = 15 + 2 = 17$$ (b) Find $$k$$ if $$b = 10$$. Substitute $$b = 10$$: $$k = 3(10) + 2 = 30 + 2 = 32$$ (c) Find $$k$$ if $$b = 23$$. Substitute $$b = 23$$: $$k = 3(23) + 2 = 69 + 2 = 71$$ --- Example 16: Given $$Q = w(4 + t)$$ (a) Find $$Q$$ if $$w = 10$$ and $$t = 3$$. Substitute $$w = 10$$ and $$t = 3$$: $$Q = 10(4 + 3) = 10(7) = 70$$ (b) Find $$t$$ if $$Q = 42$$ and $$w = 6$$. Start with: $$Q = w(4 + t)$$ Substitute known values: $$42 = 6(4 + t)$$ Divide both sides by 6: $$\frac{42}{\cancel{6}} = \cancel{6}\frac{4 + t}{6} \Rightarrow 7 = 4 + t$$ Solve for $$t$$: $$t = 7 - 4 = 3$$ (c) Find $$Q$$ if $$w = 2$$ and $$t = -5$$. Substitute: $$Q = 2(4 + (-5)) = 2(-1) = -2$$ (d) Find $$t$$ if $$Q = 45$$ and $$w = 3$$. Substitute: $$45 = 3(4 + t)$$ Divide both sides by 3: $$\frac{45}{\cancel{3}} = \cancel{3}\frac{4 + t}{3} \Rightarrow 15 = 4 + t$$ Solve for $$t$$: $$t = 15 - 4 = 11$$ --- Example 17: Total pay formula $$T = 25h + 50$$ Given $$T = 225$$, find $$h$$. Substitute: $$225 = 25h + 50$$ Subtract 50 from both sides: $$225 - 50 = 25h + 50 - 50 \Rightarrow 175 = 25h$$ Divide both sides by 25: $$\frac{175}{\cancel{25}} = \cancel{25}\frac{h}{25} \Rightarrow 7 = h$$ So, $$h = 7$$ hours. --- Example 18: Taxi cost formula $$C = 2k + 5$$ Given $$C = 21$$, find $$k$$. Substitute: $$21 = 2k + 5$$ Subtract 5 from both sides: $$21 - 5 = 2k + 5 - 5 \Rightarrow 16 = 2k$$ Divide both sides by 2: $$\frac{16}{\cancel{2}} = \cancel{2}\frac{k}{2} \Rightarrow 8 = k$$ So, $$k = 8$$ kilometres. --- Final answers: Example 15: (a) $$k=17$$, (b) $$k=32$$, (c) $$k=71$$ Example 16: (a) $$Q=70$$, (b) $$t=3$$, (c) $$Q=-2$$, (d) $$t=11$$ Example 17: $$h=7$$ hours Example 18: $$k=8$$ kilometres