Set up a pair of simultaneous equations from a description

Price of one patch in dollars

\(\displaystyle x\)

Price of one zipper pull in dollars

\(\displaystyle y\)

A craft stall sells patches and zipper pulls at fixed prices. Columns 1 and 2 show the two orders; Pulls means zipper pulls. Which pair describes the orders?

A stall sells red wristbands for five dollars each and blue wristbands for two dollars each. The total count is fixed. Move the red count to four, then to twelve. Compare the takings

What were the takings at the two requested settings, in that order?

Give each condition its own equation

First name the unknown quantities. Keep each letter's meaning fixed. Decide what each condition counts or measures before writing the pair

1. 1Two purchases When the unknowns are unit prices, each purchase has its own item counts and total. Here the letters name the two unit prices and the coefficients name the purchased counts \(\displaystyle \begin{aligned}ax+by&=P\\cx+dy&=Q\end{aligned}\)
2. 2Count and value When the unknowns are item counts, the first equation adds the counts. The second weights them by their unit prices. Here the capital letters name the total count and total value \(\displaystyle \begin{aligned}x+y&=N\\px+qy&=V\end{aligned}\)
3. 3A stated difference Subtract the smaller quantity from the larger. Here the first quantity is greater by the stated difference \(\displaystyle x-y=D\)
4. 4A future age total Both people age by the elapsed time. Here the letters name current ages, elapsed years and the future total \(\displaystyle (x+t)+(y+t)=S\)

Small pennants sold

\(\displaystyle s\)

Large pennants sold

\(\displaystyle l\)

Total pennants

\(\displaystyle 29\)

Small pennant price

\(\displaystyle \$3\)

Large pennant price

\(\displaystyle \$7\)

Takings

\(\displaystyle \$139\)

A club sold two sizes of pennant. Which pair matches the sales record?

Older cousin's current age in years

\(\displaystyle x\)

Younger cousin's current age in years

\(\displaystyle y\)

Current-age condition

\(\displaystyle x-y=9\)

Elapsed time

\(\displaystyle 6\text{ years}\)

Combined age after that time

\(\displaystyle 69\text{ years}\)

Two cousins compare their ages. Complete the constant in the equation for their current total

Both cousins get older during the elapsed time

First friend's current age in years

\(\displaystyle a\)

Second friend's current age in years

\(\displaystyle b\)

First friend is older by

\(\displaystyle 7\text{ years}\)

Elapsed time

\(\displaystyle 5\text{ years}\)

Combined age after that time

\(\displaystyle 53\text{ years}\)

Which statement identifies the error in this attempt to model two friends' ages?