{{ toc.signature }}
{{ 'ml-toc-proceed-mlc' | message }}
{{ 'ml-toc-proceed-tbs' | message }}
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}.

{{ article.displayTitle }}

{{ article.intro.summary }}
{{ ability.description }}
Lesson Settings & Tools
 {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} {{ 'ml-lesson-time-estimation' | message }}
In this lesson, systems of equations will be used to model and solve real-life situations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

How Many Cars and Motorcycles?

Mark has just started working with his father at a car dealership. They sell cars and motorcycles.

Mark counted vehicles and tires on the lot. Without considering spare tires, use this information to find the number of cars and motorcycles.
Example

Extra Dollar

Zain volunteered to work at the reception desk at the school concert.

At the end of the evening, they picked up the cash box and noticed a dollar lying on the floor next to it. Zain wonders whether the dollar belongs inside the cash box or not.

The price of tickets for the concert was one ticket for for individuals, or two tickets for for couples. Zain looked inside the cash box and found and ticket stubs for the people in attendance. Does the dollar belong inside the cash box?

Hint

Without considering the extra dollar found on the floor, write and solve a system of equations. Does the solution make sense in this context?

Solution

To find whether the dollar belongs inside the cash box, a system of equations will be written and solved. Let and be the number of tickets sold to individuals and to couples, respectively. Considering that tickets were sold, an equation can be written.
Next, since each individual's ticket is the total amount of money made by selling tickets to individuals is Similarly, since each ticket sold to couples is the total amount of money made by selling tickets to couples is Considering that the cash box contains total, a second equation can be written.
These two equations can be written together to form a system of equations.
Finally, the above system will be solved. For simplicity, the Substitution Method will be used.
Solve for
It has been found that Now, this value will be substituted into the first equation to find the value of
The solution to the system is and Considering the context, this means that without the extra dollar, tickets were sold to couples and tickets were sold to individuals. However, this is not possible, since couples tickets were only sold in pairs. Therefore, the dollar found on the floor should be placed inside the cash box.

Extra

The answer can also be found by solving a system considering the extra dollar.
The Substitution Method will be used.
Solve by substitution
This means that, considering the extra dollar, tickets were sold to couples and tickets were sold to individuals. In the context of the situation, this is a perfectly valid answer.
Example

Old and New Pennies

Pennies used to be made of copper. In the cost of this metal significantly increased, so the US government changed the composition of the coin. Since pennies have been made of zinc and plated with a thin copper coating.
Pennies made in and after weigh grams, while pennies made before weigh grams. Knowing that a roll of pennies contains coins and weighs grams, how many of the older, heavier pennies and newer, lighter pennies does this roll contain?

Hint

Write and solve a system of equations.

Solution

To find out how many of the older, heavier pennies and how many of the newer, lighter pennies this roll contains, a system of equations will be written and solved. Let be the number of older pennies and the number of newer pennies in the roll. Since the roll contains coins, an equation can be written.
Next, since older pennies weigh grams, the total weight of the older pennies in the roll is Similarly, since a newer penny weighs grams, the total weight of the newer pennies is With this information, and knowing that the coin roll weighs grams, a second equation can be written.
These two linear equations form a system of linear equations.
For simplicity the system will be solved by the Elimination Method. To do so, the first equation will be multiplied by in order for the variable to have the same coefficient in both equations.
Simplify
Now, Equation (I) will be subtracted from Equation (II).
Solve for
It has been found that Finally, this value will be substituted in the first equation to find the value
Solve for
The solution to the system is and In the context of the situation, this means that in the coin roll there are older, heavier pennies and newer, lighter pennies.
Example

Orange and Peach Juice

Tiffaniqua is selling juice to make some money for a trip to the beach. To prepare a big jug, she used oranges and peaches.

Tiffaniqua's math teacher stopped by to buy some juice and told her the amount of carbohydrates that oranges and peaches have.

Then, the teacher asked Tiffaniqua how many oranges and peaches were used in the jug of juice. Tiffaniqua decided to quiz her teacher by telling her that she used a total of fruits and that the jug contains grams of carbohydrates. How many oranges and peaches did she use?

Hint

Write and solve a system of equations.

Solution

To find out how many oranges and peaches Tiffaniqua used for a jug, a system of equations will be written and solved. Let and be the number of oranges and peaches, respectively. A first equation can be written knowing that Tiffaniqua used fruits.
Next, consider the fact that the jug contains grams of carbohydrates. It is also known that an average orange has about grams of carbohydrates and that an average peach has about grams of carbohydrates. With all this information, a second equation can be written.
These two equations form a system of linear equations.
For simplicity, the system will be solved by graphing. To do so, both linear equations must first be written in slope-intercept form.
Write in slope-intercept form
Now that both equations are written in slope-intercept form, their slopes and intercepts can be used to draw the lines on the same coordinate plane.

Finally, the point of intersection can be determined.

The lines intersect at the point with coordinates Therefore, the solution to the system is and In the context of the situation, this means that Tiffaniqua used oranges and peaches to prepare a jug of juice.

Example

Adventures on the Water is a company that organizes river safaris in Sri Lanka. They take tourists on boats along a river in the middle of the jungle for a full day.

Each boat can hold at most people and can only carry pounds of weight, including passengers and gear, for safety reasons. The company assumes that, on average, an adult weighs pounds and a child weighs pounds. It is also assumed that each group will require pounds of gear plus pounds of gear per person. There are three groups who wish to take a river safari.

• Group A has adults and children.
• Group B has adults and children.
Which of the groups, if any, can safely take a river safari?

Hint

Write and solve a system of inequalities.

Solution

To determine which of the groups can safely take a safari, a system of linear inequalities will be written and solved. Let be the number of adults and the number of children. Knowing that each boat can hold at most people, the first inequality can be written.
Next, a second inequality will be written. The company assumes that an adult weighs and a child weighs pounds. Also, for each person there are considered extra pounds of gear.
Furthermore, each group requires pounds of gear, no matter the number of people. Knowing that each boat can only carry the second inequality can be written.
The two inequalities obtained form a system of inequalities.
The above system will be solved by graphing. To do so, the equations of the boundary lines will be written in slope-intercept form. To obtain the equations of the boundaries lines, the inequality symbols will be replaced with equals signs.
Write in slope-intercept form
Now both lines can be drawn on the same coordinate plane. Since the number of adults and children cannot be negative, only the first quadrant will be considered. Also, because neither inequality is strict, the lines will be solid.
Next, a point not on the boundary lines will be tested to determine the region to be shaded. For simplicity, will be used. To start, this point will be tested in the first inequality. If a true statement is obtained, the region containing the point will be shaded. Otherwise, the opposite region will be shaded.
Since a true statement was obtained, the region that includes this point will be shaded.

By following the same procedure, the region that corresponds to the second inequality can be determined.

Test Point:
Inequality Substitute Simplify

The second inequality is also satisfied by Therefore, the region that contains this point will be shaded.

To fully see the region that satisfies both inequalities, the unwanted regiones will be removed.

Finally, a point that represents each of the three groups will be plotted to see if they belong to the shaded area.

These points will be plotted on the coordinate plane.

The points that represent groups A and B are in the shaded area, and the point that represents group C is not in the shaded area. Therefore, A and B are the only groups that can safely take a river safari.

Closure

How Many Cars and Motorcycles?

The challenge presented at the beginning of the lesson can also be modeled by a system of equations. It is known that Mark and his father sell cars and motorcycles.

In the agency, Mark counted vehicles and tires. Without considering spare tires, use this information to find the number of cars and motorcycles.

Hint

Write and solve a system of equations.

Solution

To find the number of a cars and motorcycles in the agency, a system of linear equations will be written and solved. Let be the number of cars and the number of motorcycles. Because there are vehicles, the sum of the two variables equals
Next, since spare tires are not considered, each car has tires and each motorcycle has tires. Knowing that Mark counted tires, a second equation can be written.
These two linear equations form a system.
Finally, the system will be solved. For simplicity, the Substitution Method will be used.
Solve by substitution
The solution to the system is and In this context, it means that there are cars and motorcycles in the agency.