Create Systems of Equations from Word Problems Practice AK

1. They stayed in Rome 4 nights and Paris 3 nights.

$$\require{cancel}\eqalign{p+r&=7 \qquad \text{First equation: total nights}\\85p+158r&=887 \qquad \text{Second equation: each night times cost for total cost}\\-85(p+r&=7) \qquad \text{Multiply top equation by -85}\\85p+158r&=887\\\cancel{-85p}+-85r&=595\\\cancel{85p}+158r&=887\\\hline 73r&=292\\r&=4\\p+4&=7\qquad \text{Plug r in to find p}\\p&=3}$$

2. 7 seniors went on the trip.

$$\eqalign{a+s&=87 \qquad \text{First equation: total people}\\42a+32s&=3584 \qquad \text{Second equation: each whale watcher times cost for total cost}\\-42(a+s&=78) \qquad \text{Multiply top equation by -42}\\42a+32sr&=3584\\\cancel{-42a}+-42s&=3654\\\cancel{42a}+32s&=3584\\\hline -10s&=-70\\s&=7}$$

3. 12 peppers.  8 tomatoes.

$$\eqalign{p+t&=20 \qquad \text{First equation: total plants}\\.45p+.3t&=7.80 \qquad \text{Second equation: each plant times cost for total cost}\\-.3(p+t&=20) \quad \text{Multiply top equation by -.3}\\.45p+.3t&=7.80\\-.3p+\cancel{-.3t}&=6\\.45p+\cancel{.3t}&=7.80\\\hline .15p&=1.80\\p=12\\p+t&=20\\12+t&=20\\t&=8}$$

4. 8 quarterly subscriptions.

$$\eqalign{a+q&=42 \qquad \text{First equation: total subscriptions}\\14a+8q&=540 \qquad \text{Second equation: each cost times type for total cost}\\-8(a+q&=42) \quad \text{Multiply top equation by -8}\\14a+8q&=540\\-8a+\cancel{-8q}&=-336\\14a+\cancel{8a}&=540\\\hline 6a&=204\\a&=34\\a+q&=42\\34+q&=42 \qquad \text{Plug a in to find q}\\q&=8}$$

5. 14 projects.

$$\eqalign{h+p&=30 \qquad \text{First equation: total assingments}\\10h+40p&=720 \qquad \text{Second equation: points times assignments for total points}\\-10(h+p&=30) \quad \text{Multiply top equation by -10}\\10h+40p&=720\\\cancel{-10h}+-10p&=-300\\\cancel{10h}+40p&=720\\\hline 30p&=420\\p&=14}$$