Question 126·Hard·Systems of Two Linear Equations in Two Variables
Pack A contains 3 red marbles and 2 blue marbles, and Pack B contains 5 red marbles and 4 blue marbles. A gift shop purchases a total of 210 red marbles and 160 blue marbles by buying some of Pack A and some of Pack B. How many of Pack B does the shop purchase?
For systems-of-equations word problems, start by clearly defining variables for the unknown quantities, then write one equation for each distinct condition in the problem (here, total red marbles and total blue marbles). Check whether you can simplify one equation, then use substitution or elimination—whichever keeps numbers smaller. Always finish by answering the specific question asked (which variable or expression they want) and quickly plug your solution back into both original equations to confirm it satisfies all given totals.
Hints
Turn the word problem into algebra
Assign a variable to the number of Pack A and another variable to the number of Pack B. How can you write equations using the total numbers of red and blue marbles?
Write separate equations for red and blue marbles
Use the counts in each pack: Pack A has 3 red and 2 blue; Pack B has 5 red and 4 blue. Use these to form one equation for red marbles totaling 210 and another for blue marbles totaling 160.
Simplify and choose a solving method
After you have the two equations, see if one can be simplified (for example by dividing all terms). Then decide whether substitution or elimination looks quicker.
Focus on the variable you need
The problem asks for the number of Pack B. As you solve the system, keep track of which variable represents Pack B and make sure you actually solve for that one at the end.
Desmos Guide
Enter the system of equations
In Desmos, type the two equations as
3x + 5y = 2102x + 4y = 160so that both lines are graphed on the coordinate plane.
Find the intersection point
Zoom or pan until you can see where the two lines cross, then tap or click on the intersection point. Desmos will display the coordinates of this point.
Read the value that represents Pack B
Remember that represents the number of Pack A and represents the number of Pack B. Use the ‑coordinate of the intersection point as the number of Pack B the shop buys.
Step-by-step Explanation
Define variables and write equations
Let be the number of Pack A the shop buys, and let be the number of Pack B.
From the red marbles:
- Each Pack A has 3 red marbles, so Pack A contributes red marbles.
- Each Pack B has 5 red marbles, so Pack B contributes red marbles.
- The total red marbles is 210, so we get the equation
From the blue marbles:
- Each Pack A has 2 blue marbles, so Pack A contributes blue marbles.
- Each Pack B has 4 blue marbles, so Pack B contributes blue marbles.
- The total blue marbles is 160, so we get the equation
Now we have a system of two equations in and . The question asks for , the number of Pack B.
Simplify one of the equations
Take the blue-marble equation
and divide every term by 2 to make it simpler:
Now our system is
Use substitution to solve the system
From the simplified blue equation
solve for :
Substitute this expression for into the red equation :
Now combine like terms.
Finish the algebra and answer the question
Continuing from the previous step:
Subtract 240 from both sides:
Multiply both sides by :
So the shop buys of Pack B, which corresponds to answer choice C.