Question 46·Easy·Systems of Two Linear Equations in Two Variables
Maria and Luis bought notebooks and folders at the same store. Each notebook cost the same amount, and each folder cost the same amount.
• Maria bought 2 notebooks and 3 folders for a total of $19. • Luis bought 5 notebooks and 2 folders for a total of $31.
Which of the following systems of linear equations represents this situation, where is the price, in dollars, of one notebook and is the price, in dollars, of one folder?
For word problems that ask for a system of equations, start by clearly identifying what each variable represents, then translate each sentence into an equation using the pattern (number of each item)·(price of that item) = total. Write one equation per person or per scenario, making sure the coefficients match the counts and the constant term matches the total. Finally, scan the answer choices for the option that contains both of your equations; you do not need to solve the system unless you want to double-check your work.
Hints
Focus on what x and y represent
The problem defines and for you. Think about how you would write the total cost of some notebooks and folders using and .
Turn Maria’s description into an equation
Maria buys 2 notebooks and 3 folders for 19 dollars. Using the form (number of notebooks) + (number of folders) = total, what does her equation look like?
Turn Luis’s description into an equation
Do the same for Luis: he buys 5 notebooks and 2 folders for 31 dollars. Once you have both equations, look for the answer choice that contains both of them together.
Desmos Guide
Enter the equations from one answer choice
Pick one answer choice. In Desmos, type the first equation from that choice on one line (for example, something like 2x + 3y = 19), and the second equation from the same choice on the next line. Desmos will graph both lines at once.
Find the intersection point of the two lines
Look for the point where the two lines intersect. Click or tap that intersection; Desmos will show its coordinates . Interpret this as price of one notebook and price of one folder for that answer choice.
Check whether those prices match both people’s totals
Using the intersection values for and , compute the total cost for Maria (2 notebooks and 3 folders) and for Luis (5 notebooks and 2 folders), either by hand or by entering expressions like 2x + 3y and 5x + 2y using those values. If you do not get 19 for Maria and 31 for Luis, that answer choice is wrong; try another choice until you find a pair of equations whose solution gives both correct totals.
Step-by-step Explanation
Define the variables and the equation pattern
The problem tells you that is the price of one notebook and is the price of one folder. Any total cost will be:
We will use this pattern for Maria and for Luis.
Write the equation for Maria’s purchase
Maria bought 2 notebooks and 3 folders for a total of 19 dollars.
Using the pattern:
- Number of notebooks: 2, so the notebook term is .
- Number of folders: 3, so the folder term is .
- Total cost: 19.
So Maria’s equation is:
Write the equation for Luis’s purchase
Luis bought 5 notebooks and 2 folders for a total of 31 dollars.
Using the same pattern:
- Number of notebooks: 5, so the notebook term is .
- Number of folders: 2, so the folder term is .
- Total cost: 31.
So Luis’s equation is:
Match both equations to the answer choices
We now know the system that represents the situation must be
Looking at the options, this pair appears in choice A as , , so choice A is the correct answer.