Sam collects insects. However, he only
collects ants and beetles. One night while
collecting insects for his collection, Sam had
an unproductive hunt. The number of ants
plus twice the number of beetles was only
eight. The difference between the number of
ants and beetles was -4. How many ants and
how many beetles did he catch?
a + 2b = 8
a - b = -4
Multiply the second equation by -1.
-a + b = 4
Add the two equations:
3b = 12
b = 4
Put 4 into one equation:
a + 2(4) = 8
a + 8 = 8
a = 0
He caught 0 ants and 4 beetles.
To solve this problem, we can break it down into two equations and use algebra to find the values of ants and beetles that satisfy both equations.
Let's represent the number of ants as 'A' and the number of beetles as 'B'.
From the information given, we can form two equations:
Equation 1: "The number of ants plus twice the number of beetles was only eight."
This can be written as: A + 2B = 8.
Equation 2: "The difference between the number of ants and beetles was -4."
This can be written as: A - B = -4.
Now we have a system of two equations:
Equation 1: A + 2B = 8
Equation 2: A - B = -4
To solve this system, we can use the method of substitution or elimination.
Let's use substitution:
From Equation 2, we can express A in terms of B by adding B to both sides:
A = B - 4.
Substitute this expression for A into Equation 1:
B - 4 + 2B = 8.
Combine like terms:
3B - 4 = 8.
Add 4 to both sides:
3B = 12.
Divide both sides by 3:
B = 4.
Now that we have the value of B, we can substitute it back into the equation to find A:
A = B - 4 = 4 - 4 = 0.
So, Sam caught 0 ants and 4 beetles.