QuestionA fruit stand has to decide what to charge for their produce. They need for 4 apples and 4 oranges. They also need for 6 apples and 6 oranges. We put this information into a system of linear equations.
Can we find a unique price for an apple and an orange?
Studdy Solution
STEP 1
What is this asking? If 4 apples and 4 oranges cost , and 6 apples and 6 oranges cost , can we figure out the individual price of an apple and an orange? Watch out! We might be tempted to assume we have enough information just because we have two equations, but we need to be careful and check if they're truly different!
STEP 2
1. Set up the equations
2. Simplify the equations
3. Analyze for solutions
STEP 3
Let's use for the price of an apple and for the price of an orange.
Our first equation represents "4 apples and 4 oranges cost ":
STEP 4
Our second equation represents "6 apples and 6 oranges cost ":
STEP 5
Let's simplify the first equation by dividing both sides by their **greatest common divisor**, which is **4**.
Remember, we're dividing both sides by 4 to keep the equation balanced!
This gives us:
So, one apple and one orange cost .
STEP 6
Now, let's simplify the second equation.
The **greatest common divisor** of 6, 6, and 12 is **6**.
Dividing both sides by **6** gives us:
So, one apple and one orange cost .
STEP 7
Uh oh!
We have a problem!
One apple plus one orange apparently costs both and .
That doesn't make sense!
STEP 8
This tells us the information we were given is contradictory.
We can't find a single price for an apple and an orange that satisfies both conditions.
Our equations are inconsistent!
STEP 9
We cannot find a unique price for an apple and an orange.
The given information leads to contradictory results.
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