# Applications of Linear Equations

You will need scratch paper to do this lesson.

## Balance problems

You have a balanced scale. On one side of the scale are three equally heavy sacks of barley and a 1-pound weight, and on the other side is a 10-pound weight. Since the scale balances, the two sides weigh the same amount. If \$b\$ represents the weight of a sack of barley in pounds, the objects on this scale obey the equation \$3b+1=10\$.

Solve this equation to find the weight of a single sack of barley, and check your solution.

 A scale balances when it has four equally heavy bags of sugar and a 5-pound weight on one side, and a 25-pound weight on the other side. If \$s\$ represents the weight of a bag of sugar in pounds, what equation do the objects on this scale obey?

Solve the equation you found in scaleqn, and check your solution.

Each row of the table below lists some weights and equally-heavy objects that are on a balanced scale. Write down an equation that the objects on the scale obey. Then solve it and check your solution by using scratch paper or the grids to the left.

## Budgeting

The senior class of Mathscribe High is planning a party. They have \\$500 that they can spend. It costs \\$100 to rent the school gym for the party, and they can buy bottles of soda for \\$2 each. They want to spend all their money on soda except what it costs to rent the gym. If \$s\$ represents the number of bottles of soda they can buy, an equation for \$s\$ is \$2s+100=500\$.

By solving this equation, find the number of bottles of soda the seniors can buy. Check your solution.

 If the seniors have \\$600 they can spend and the gym still costs \\$100, what is an equation for the number of bottles of soda (\$s\$) they can buy?
 Solve this equation using scratch paper.
 If the seniors still have \\$600 they can spend, but the price of the gym is reduced to \\$60, what is an equation for the number of bottles of soda they can buy?
 Solve this equation using scratch paper.
 If the seniors still have \\$600 to spend and the gym still costs \\$60, but the price of soda goes up to \\$3 per bottle, what is an equation for the number of bottles of soda they can buy?
 Solve this equation using scratch paper.

## Temperature conversion

There are two common ways to measure temperature: in degrees Celsius, and in degrees Fahrenheit. If \$C\$ represents the current temperature in degrees Celsius and \$F\$ represents the current temperature in degrees Fahrenheit, then you can convert between them with the equation:

\$\$ 9/5 C + 32 = F \$\$

Convert each temperature in the table below from degrees Celsius into degrees Fahrenheit.

Temperature
(degrees Celsius)
Temperature
(degrees Fahrenheit)

If the current temperature is 68 degrees Fahrenheit, an equation for the current temperature in degrees Celsius is \$\$9/5 C + 32 = 68\$\$. Solve this equation, and check your solution.

Each row of this table gives a temperature in degrees Fahrenheit. Write an equation for the corresponding temperature in degrees Celsius, and then solve it as you did in FtoCqn. Use scratch paper.

Temperature
(degrees Fahrenheit)
Temperature
(degrees Celsius)