Question 80·Medium·Systems of Two Linear Equations in Two Variables
Leah sold cupcakes for $2 each and cookies for $1 each. If she sold a total of 50 items and collected $80, how many cupcakes did she sell?
For systems of equations word problems, first define clear variables (like cupcakes and cookies), then write one equation for the item count and another for the total cost. Put both equations in standard form and use elimination to quickly remove one variable—subtraction is often fastest when the coefficients already match—then solve for the remaining variable and check it against the original conditions before choosing your answer.
Hints
Assign variables
Let one variable represent the number of cupcakes and another represent the number of cookies. What should each variable stand for?
Create equations from totals
Use the total number of items (50) to write one equation, and the total money collected ($80) to write another. Each item’s price should appear in the money equation.
Choose a solving method
You now have two equations with two variables. Try eliminating one variable by subtracting one equation from the other, or use substitution if you prefer.
Check your solution
After you find the number of cupcakes, plug both numbers back into the original conditions: total items should be 50 and total money should be $80.
Desmos Guide
Enter the equations
In Desmos, type c + k = 50 on one line and 2c + k = 80 on another line. Desmos will treat c and k as the horizontal and vertical axes variables.
Find the intersection point
Look for the point where the two lines intersect. The -coordinate (the x-value) of this intersection represents the number of cupcakes, and the -coordinate (the y-value) represents the number of cookies.
Interpret the result
Read the -coordinate of the intersection point from Desmos and match that value to the closest choice for the number of cupcakes in the answer options.
Step-by-step Explanation
Define variables
Let be the number of cupcakes Leah sold and be the number of cookies she sold.
We are asked to find the value of .
Translate the word problem into equations
Use the information in the problem to write equations:
- Total number of items: she sold 50 items in all.
- So, .
- Total money collected: each cupcake is $2 and each cookie is $1, and she collected $80 in total.
- So, , or simply .
Set up elimination
We have the system:
Subtract the first equation from the second to eliminate :
This eliminates and isolates ; solve this equation for .
Answer the question
We find that , meaning Leah sold 30 cupcakes.
So, the correct answer is 30.