3.3 Homework: simultaneous equations

Years after their famous race, the tortoise and the hare agree on a re-match. As they begin the race, the tortoise plods along at $2\frac{1}{2}$ mph, slow but confident. The hare zips out at $8\frac{1}{2}$ mph, determined not to repeat his original mistake... and he doesn't! The hare never slows down, and reaches the finish line 45 minutes (that is, $\frac{3}{4}$ of an hour) before the tortoise.

• a Did they run the same amount of time as each other?
• b Did hey run the same distance?
• c Clearly define and label the two variables in this problem. Note that your answers to (a) and (b) will have a lot to do with how you do this?
• d Based on your variables, write the equation $d=rt$ for the tortoise.
• e Based on your variables, write the equation $d=rt$ for the hare.
• f Now answer the question: how long did the tortoise run?

$\mathrm{6x}+\mathrm{2y}=6$

$x-y=5$

• a Solve by substitution.
• b Check your answers.

$\mathrm{3x}+\mathrm{2y}=\text{26}$

$\mathrm{2x}-\mathrm{4y}=4$

• a Solve by substitution.
• b Check your answers.

$\mathrm{2y}-x=4$

$\mathrm{2x}+\text{20}y=4$

• a Solve by substitution.
• b Check your answers.

$7x-3=y$

$14x=2y+6$

• a Solve by substitution.
• b Check your answers.

$\mathrm{3x}+\mathrm{4y}=\text{12}$

$\mathrm{5x}-\mathrm{3y}=\text{20}$

• a Solve any way you like.
• b Check your answers.

$\text{ax}+y=6$

$\mathrm{2x}+y=4$

• a Solve any way you like. You are solving for $x$ and $y$ ; $a$ is just a constant. (So your final answer will say “ $x$ = blah-blah , $y$ = blah-blah .” The blah-blah will both have $a$ in them.)
• b Check your answers.