Activity
To get snacks for a class trip, Clare went to the “bulk” section of the grocery store, where she could buy any quantity of a product and the prices are usually good.
Clare purchased some salted almonds at $6 a pound and some dried figs at $9 per pound. She spent $75 before tax.
If she bought 2 pounds of almonds, how many pounds of figs did she buy?
7
If she bought 1 pound of figs, how many pounds of almonds did she buy?
11
Write an equation that describes the relationship between pounds of figs and pounds of almonds that Clare bought and the dollar amount that she paid. Be sure to specify what the variables represent.
Compare your answer:
: pounds of almonds, : pounds of figs
For questions 4 and 5, use the graph that represents the quantities in the situation.
Choose any point on the line, state its coordinates, and explain what it tells us.
Compare your answer:
Your answer may vary, but here is a sample. Point A or (2, 7). Clare bought 2 pounds of almonds and 7 pounds of figs for a total of $75.
Choose any point that is not on the line, state its coordinates, and explain what it tells us.
Compare your answer:
Your answer may vary, but here is a sample. Point D or . Clare bought 1 pound of almonds and 1 pound of figs, and the total was not $75.
Video: Using a Graph to Solve an Equation
Watch the following video to further explore the above scenario.
Self Check
Which point is a solution to the line graphed below?
Additional Resources
Determining the Meaning of Solutions of a Graphed Line
Suppose we are buying beans and rice to feed a large gathering of people, and we plan to spend $120 on the two ingredients. Beans cost $2 a pound, and rice costs $0.50 a pound.
If represents pounds of beans and pounds of rice, the equation can represent the constraints in this situation.
The graph of shows a straight line.
Each point on the line is a pair of - and -values that make the equation true and is thus a solution. It is also a pair of values that satisfy the constraints in the situation.
- The point is on the line. If we buy 10 pounds of beans and 200 pounds of rice, the cost will be , which equals 120.
- The points and are also on the line. If we buy only beans—60 pounds of them—and no rice, we will spend $120. If we buy 45 pounds of beans and 60 pounds of rice, we will also spend $120.
What about points that are not on the line? They are not solutions because they don’t satisfy the constraints, but they still have meaning in the situation.
- The point is not on the line. Buying 20 pounds of beans and 80 pounds of rice costs or 80, which does not equal 120. This combination costs less than what we intend to spend.
- The point means that we buy 70 pounds of beans and 180 pounds of rice. It will cost or 230, which is over our budget of 120.
Let’s look at an example.
Example
1. Looking at the graph about purchasing beans and rice, is the point (30, 120) a solution?
Yes, because .
2. What does the point (30, 120) mean in this situation?
Compare your answer:
The point means that 30 pounds of beans and 120 pounds of rice were purchased. Since the point is on the line, the total spent was $120.
Try it
Try It: Determining the Meaning of Solutions of a Graphed Line
At a high school baseball game, the concession stand sold hot dogs and hamburgers. Hot dogs were $1.50, and hamburgers cost $3. The goal is to make $150 for the game.
What is the meaning of the point on the graph?
Compare your answer:
Here is how to determine the meaning of the point on this graph.
The -axis is labeled “number of hot dogs,” and the -axis is labeled “number of hamburgers.” Since , there were 40 hot dogs sold, and since , there were 30 hamburgers sold. Since the point is on the line, the total amount from the hot dogs and hamburgers sold was $150.