24. At a vegetable stand, you bought 3 pounds of peppers for [tex]\$4.50[/tex]. Green peppers cost [tex]\$1[/tex] per pound and orange peppers cost [tex]\$4[/tex] per pound. Find how many pounds of each kind of peppers you bought.



Answer :

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.

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