Chapter 14: Review

The chapter problem, solved

Let $f_A$ be Company A's fixed fee and $r_A$ its price per kilometre. The two fares give

Subtract the first equation from the second: $4r_A=8$, so $r_A=2$. Then $f_A+4(2)=14$, so $f_A=6$.

For Company B, let $f_B$ be its fixed fee and $r_B$ its price per kilometre.

Subtracting gives $4r_B=8$, so $r_B=2$. Then $f_B+4(2)=16$, so $f_B=8$.

Both companies charge £2 per kilometre. Company A has a £6 fixed fee; Company B has an £8 fixed fee. This confirms the opening prediction can be checked by comparing the rates.

Mixed practice: choose the method and explain why

For each question, name a method before working. Give a reason for your choice.

1. Solve $x+y=12$ and $x-y=4$.
2. Solve $y=3x-2$ and $x+y=10$.
3. Find the intersection of $y=x+1$ and $y=-2x+7$.
4. Solve $y=x+2$ and $y=x^2-2x-3$.
5. Use $x_{n+1}=\frac12(x_n+6/x_n)$ with $x_0=2$ to find $x_3$.
6. Work out $3(2.5-0.7)$, then explain how inverse operations could check it.
7. A temperature rises from $-4^\circ$C by $9^\circ$C. State the new temperature.
8. Solve $2x+3=15$. Then use your value as the $x$ coordinate in $y=2x-1$.

What you can now do

• choose elimination, substitution, or a graph for a pair of linear equations;
• solve linear and quadratic simultaneous equations and check each pair;
• follow an iteration rule and report the requested approximation.

Your relevance reflection

Return to your example from the chapter opener. What were the two unknowns, and what information would let you find them? Has a method from this chapter helped you describe the problem more precisely?