To solve this problem, we need to set up a system of equations that represents the given information. Let's denote:
- [tex]\( x \)[/tex] as the number of pounds of green peppers.
- [tex]\( y \)[/tex] as the number of pounds of orange peppers.
We know the following:
1. The total weight of the peppers is 3 pounds.
2. The total cost of the peppers is [tex]$4.50.
3. Green peppers cost $[/tex]1 per pound.
4. Orange peppers cost $4 per pound.
From the problem's constraints, we can set up the following equations:
Equation 1: Total weight of the peppers
[tex]\[ x + y = 3 \][/tex]
Equation 2: Total cost of the peppers
[tex]\[ 1 \cdot x + 4 \cdot y = 4.50 \][/tex]
[tex]\[ x + 4y = 4.50 \][/tex]
Now we have a system of linear equations:
[tex]\[ \begin{cases}
x + y = 3 \\
x + 4y = 4.50
\end{cases} \][/tex]
We can solve this system using the substitution or elimination method. In this case, we'll use the subtraction method to eliminate [tex]\( x \)[/tex]:
Subtract the first equation from the second equation:
[tex]\[ (x + 4y) - (x + y) = 4.50 - 3 \][/tex]
[tex]\[ x + 4y - x - y = 1.50 \][/tex]
[tex]\[ 3y = 1.50 \][/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1.50}{3} \][/tex]
[tex]\[ y = 0.5 \][/tex]
So, the number of pounds of orange peppers ([tex]\( y \)[/tex]) is [tex]\( 0.5 \)[/tex] pounds.
Next, we use Equation 1 to find [tex]\( x \)[/tex]:
[tex]\[ x + y = 3 \][/tex]
[tex]\[ x + 0.5 = 3 \][/tex]
[tex]\[ x = 3 - 0.5 \][/tex]
[tex]\[ x = 2.5 \][/tex]
So, the number of pounds of green peppers ([tex]\( x \)[/tex]) is [tex]\( 2.5 \)[/tex] pounds.
Therefore, you bought:
- [tex]\( 2.5 \)[/tex] pounds of green peppers
- [tex]\( 0.5 \)[/tex] pounds of orange peppers.
