Question 109·Easy·Systems of Two Linear Equations in Two Variables
Let be the number of red pens and be the number of blue pens in a box. The box has 6 more blue pens than red pens, and there are 42 pens in total. Which system of equations represents this situation?
For word problems that ask you to choose a system of equations, start by clearly defining each variable in words (for example, r = number of red pens, b = number of blue pens). Then translate each key phrase step by step: phrases like "is" become an equals sign, "more than" or "less than" suggest addition or subtraction, and "in total" usually means you are adding quantities. Write short word equations first (e.g., blue = red + 6, red + blue = 42), then convert them into algebra with the variable letters and match to the answer choice that uses exactly those equations in the same relationships.
Hints
Identify what r and b stand for
Make sure you know which letter represents red pens and which represents blue pens. Which one does the problem say there are more of?
Translate "6 more blue pens than red pens"
If one quantity is 6 more than another, think about which goes on each side of the equals sign. Should the larger quantity or the smaller quantity be alone on one side?
Translate the total number of pens
The problem says there are 42 pens in total. How can you write an equation that adds the number of red pens and blue pens to equal 42?
Desmos Guide
Enter a system from one answer choice
In Desmos, type the two equations from one answer choice exactly as written (for example, you can type them using and as variables). Desmos will graph two lines that represent that system.
Find and interpret the intersection point
Look for the point where the two lines intersect. The coordinates of this point give values for the two variables (red pens and blue pens). Check whether those values make sense for the story: do they add up to 42, and is the number of blue pens 6 more than the number of red pens?
Test the other answer choices if needed
Repeat the process for the other answer choices. Only the system whose intersection point gives numbers that both add to 42 and show blue pens being 6 more than red pens correctly represents the situation.
Step-by-step Explanation
Understand the variables and the relationships in words
We are told that is the number of red pens and is the number of blue pens.
The sentence "The box has 6 more blue pens than red pens" tells us that the number of blue pens is greater than the number of red pens by 6.
The sentence "there are 42 pens in total" tells us that if we add the number of red and blue pens, the sum is 42.
Turn each sentence into a word equation
First relationship (comparison):
- Number of blue pens = number of red pens + 6.
- This matches the pattern "larger amount = smaller amount + 6".
Second relationship (total):
- Number of red pens + number of blue pens = 42.
Now we have two equations described in words; the next step is to write them using and and match them to an answer choice.
Write the algebraic system and match the answer choice
Translate the word equations into algebra:
- Number of blue pens = number of red pens + 6 becomes .
- Number of red pens + number of blue pens = 42 becomes .
So the system that represents the situation is
This matches answer choice A.