Question 1·Easy·Systems of Two Linear Equations in Two Variables
A craft-fair vendor sells small candles for $4 each and large candles for $7 each. In one day, the vendor sold a total of 40 candles and collected $214 in sales.
If represents the number of small candles sold and represents the number of large candles sold, which of the following systems of equations can be solved to find and ?
For word problems asking which system of equations models a situation, always start by writing down what each variable represents in words. Then create one equation for a simple count or total (like total items or total hours) and a second equation for money, distance, or another quantity that uses prices or rates as coefficients. Make sure the units match on each side: the equation with prices should equal the total money, and the equation that just adds variables should equal the total number of items. Finally, compare your equations to each answer choice and pick the one that matches exactly, checking that each coefficient is attached to the correct variable.
Hints
Match equations to what is being counted
One equation should be about the number of candles and the other about the amount of money. Decide which right-hand side (40 or 214) belongs with each idea.
Build the total-candles equation
If is the number of small candles and is the number of large candles, how do you write an equation that says the total number of candles is 40?
Build the money equation carefully
The small candles are $4 each and the large candles are $7 each. Which expression combines and to give the total money collected, and what number should that total money equal?
Desmos Guide
Relabel variables for Desmos
In Desmos, use for the number of small candles and for the number of large candles (so think and ).
Test a candidate system
Pick one answer choice. Replace with and with , and type both of its equations into Desmos on separate lines (for example, something like 4x + 7y = 214 and x + y = 40).
Check if the equations match the story
For the system you typed, ask:
- Does the equation involving the prices 4 and 7 represent the total money (so its right side should be 214)?
- Does the simpler equation with just and represent the total number of candles (so its right side should be 40)?
Use the intersection only to verify counts
If you want, look at the intersection point of the two lines to see the values of that satisfy the system. Then check in the original word problem whether that pair really gives 40 candles in total and $214 when you compute ; only a correctly modeled system will pass both checks.
Step-by-step Explanation
Understand what the variables mean
The problem defines the variables for you:
- = number of small candles sold
- = number of large candles sold
Every equation you write must use and in a way that matches this meaning.
Write the equation for the number of candles
The vendor sold a total of 40 candles.
That total is made up of small candles plus large candles, so:
- number of small candles =
- number of large candles =
- total candles =
So the count equation is .
Write the equation for the total money
Each small candle costs $4 and each large candle costs $7, and the total money collected was $214.
- Money from small candles = ($4 times small candles)
- Money from large candles = ($7 times large candles)
- Total money = $214
So the money equation is .
Form the system and match it to a choice
Putting both equations together, the system that models the situation is:
- (total money)
- (total number of candles)
This matches answer choice B.