Question 88·Easy·Linear Equations in Two Variables
At a school fundraiser, bracelets cost $2 each and keychains cost $3 each. If Lena bought 7 items in total and spent $17, how many keychains did she buy?
For SAT word problems that describe buying items at different prices with a total number and total cost, immediately define variables for each item type and write two equations: one for the item count and one for the total cost. Then solve the system quickly using substitution or by plugging in the answer choices for the requested quantity and checking which one fits the money equation; this is often fastest when there are only a few simple options.
Hints
Translate the story into variables
Let one variable represent the number of bracelets and another represent the number of keychains. How can you write an equation using the fact that she bought 7 items total?
Use the prices to make a second equation
Write an expression for the total cost of the bracelets and another for the total cost of the keychains using their prices and your variables. Set their sum equal to 17.
Solve the system efficiently
You now have two equations with two variables. Try solving the first equation for one variable and substituting into the second, or use the answer choices and check which one makes the total cost $17.
Desmos Guide
Enter the system of equations
In Desmos, type b + k = 7 on one line and 2b + 3k = 17 on another. If Desmos prefers x and y, you can instead use x + y = 7 and 2x + 3y = 17.
Find the intersection point
Desmos will draw two lines. Tap or click on their intersection point; note the coordinates of this point.
Interpret the intersection
Use the coordinate that represents the number of keychains (if you used and , decide which variable you assigned to keychains). That coordinate is the number of keychains Lena bought.
Step-by-step Explanation
Define variables and use the item total
Let be the number of bracelets and be the number of keychains.
We are told Lena bought 7 items in total, so the first equation is:
Use the cost information to write a second equation
Bracelets cost $2 each, so is the total cost of bracelets.
Keychains cost $3 each, so is the total cost of keychains.
The total amount spent was $17, so the second equation is:
Now we have a system of equations:
Solve the system to find the number of keychains
From the first equation, solve for :
Substitute this into the second equation:
Simplify:
Subtract 14 from both sides:
So Lena bought 3 keychains.