Certainly! Let's go through the process step by step to determine the cost of apples and bananas.
We are given the following system of linear equations:
1. [tex]\( 3a + 4b = 1.90 \)[/tex]
2. [tex]\( 7a + 3b = 2.85 \)[/tex]
Where:
- [tex]\(a\)[/tex] is the cost of one apple.
- [tex]\(b\)[/tex] is the cost of one banana.
### Step 1: Write down the equations
From the problem, we have:
[tex]\[ 3a + 4b = 1.90 \][/tex]
[tex]\[ 7a + 3b = 2.85 \][/tex]
### Step 2: Solve the system of equations
We need to solve this system to find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
To solve these equations, we'll use a combination of elimination and substitution or use matrix methods. For this initially, let's manipulate the equations to eliminate one of the variables.
### Step 3: Multiply the equations
We can multiply the equations to make the coefficients of one of the variables match. Here, we’ll eliminate [tex]\(b\)[/tex].
Multiply the first equation by 3:
[tex]\[ 3(3a + 4b) = 3(1.90) \][/tex]
[tex]\[ 9a + 12b = 5.70 \][/tex] [tex]\(\text{[Equation 3]}\)[/tex]
Multiply the second equation by 4:
[tex]\[ 4(7a + 3b) = 4(2.85) \][/tex]
[tex]\[ 28a + 12b = 11.40 \][/tex] [tex]\(\text{[Equation 4]}\)[/tex]
### Step 4: Subtract the equations
Subtract Equation 3 from Equation 4:
[tex]\[ (28a + 12b) - (9a + 12b) = 11.40 - 5.70 \][/tex]
[tex]\[ 28a + 12b - 9a - 12b = 5.70 \][/tex]
[tex]\[ 19a = 5.70 \][/tex]
### Step 5: Solve for [tex]\( a \)[/tex]
[tex]\[ a = \frac{5.70}{19} \][/tex]
[tex]\[ a = 0.30 \][/tex]
### Step 6: Substitute back to find [tex]\( b \)[/tex]
Now that we have [tex]\(a = 0.30\)[/tex], we can substitute this back into one of the original equations to solve for [tex]\(b\)[/tex]. Let's use the first equation:
[tex]\[ 3a + 4b = 1.90 \][/tex]
[tex]\[ 3(0.30) + 4b = 1.90 \][/tex]
[tex]\[ 0.90 + 4b = 1.90 \][/tex]
[tex]\[ 4b = 1.90 - 0.90 \][/tex]
[tex]\[ 4b = 1.00 \][/tex]
[tex]\[ b = \frac{1.00}{4} \][/tex]
[tex]\[ b = 0.25 \][/tex]
### Conclusion
The cost of one apple is [tex]\( £0.30 \)[/tex] and the cost of one banana is [tex]\( £0.25 \)[/tex].
