Question 62·Medium·Systems of Two Linear Equations in Two Variables
At a farmers market, Maria bought a total of 14 pieces of fruit consisting only of apples and oranges. Apples cost $2 each and oranges cost $3 each. If Maria spent $34 in total, how many oranges did she buy?
For word problems that give a total number of items and a total cost with two item types, quickly define variables for each item type and write two equations: one for the total number of items and one for the total cost. Then use substitution or elimination to solve the system, and finally check your solution by plugging back into both equations; if answer choices are given, you can also plug each option into the cost equation to see which one matches the total cost while keeping the total number of items correct.
Hints
Translate the words into equations
Try assigning variables: let one variable represent the number of apples and another for oranges. How can you express the total number of fruits and the total cost using these variables?
Use the total number of fruits
From the fact that Maria bought 14 fruits total, write an equation that relates the number of apples and oranges.
Use the total cost
Using the prices of apples and oranges and the total cost of $34, write a second equation involving the same variables.
Solve the system
Use substitution or elimination to solve your two equations and find the value of the variable that represents the number of oranges.
Desmos Guide
Enter the two equations
In Desmos, type the two equations as they are: a + o = 14 on one line and 2a + 3o = 34 on another line. Desmos will interpret these as lines in the -plane if you use x and y instead (for example, x + y = 14 and 2y + 3x = 34).
View the intersection point
Look at the graph where the two lines intersect. Click on the intersection point; Desmos will show its coordinates , where corresponds to the number of oranges and to the number of apples.
Read the number of oranges
From the intersection point’s coordinates, identify the -value. That -value is the number of oranges that satisfies both the total fruits and total cost conditions.
Step-by-step Explanation
Define variables and write the equations
Let be the number of apples and be the number of oranges.
From the problem:
- Total pieces of fruit:
- Total cost: apples are $2 each and oranges are $3 each, so .
So we have the system:
Solve the system using substitution
From the first equation, solve for :
Substitute into the cost equation :
Simplify and solve for the number of oranges
Distribute and combine like terms:
Now solve for :
So Maria bought 6 oranges.