Let's break down the problem step by step.
Part A: Formulating the system of equations
We know from the problem that:
- Ted bought 4 shirts and 2 ties for a total of \[tex]$95.
- Stephen bought 3 shirts and 3 ties for a total of \$[/tex]84.
We need to use these statements to formulate a system of equations.
Let's denote:
- [tex]\( s \)[/tex] as the cost of one shirt in dollars.
- [tex]\( t \)[/tex] as the cost of one tie in dollars.
From Ted's purchase, we have the equation:
[tex]\[ 4s + 2t = 95 \][/tex]
From Stephen's purchase, we have the equation:
[tex]\[ 3s + 3t = 84 \][/tex]
Looking at the given options, we find that Option B matches our equations:
[tex]\[ 4s + 2t = 95 \][/tex]
[tex]\[ 3s + 3t = 84 \][/tex]
Hence, the correct answer for Part A is:
B. [tex]\( 4s + 2t = 95 \)[/tex], [tex]\( 3s + 3t = 84 \)[/tex]
Now we'll solve this system of equations to find the values of [tex]\( s \)[/tex] and [tex]\( t \)[/tex].
Part B: Solving the system of equations
Using the system of equations:
[tex]\[ 4s + 2t = 95 \][/tex]
[tex]\[ 3s + 3t = 84 \][/tex]
We solve these equations simultaneously.
From solving these equations, the solutions are:
[tex]\[ s = \frac{39}{2} \][/tex]
[tex]\[ t = \frac{17}{2} \][/tex]
This means the cost of each shirt ([tex]\(s\)[/tex]) is \[tex]$19.50 and the cost of each tie (\(t\)) is \$[/tex]8.50.
Part C: Calculating the total price for Linda's purchase
Linda bought 1 shirt and 2 ties. To find the total price, we calculate:
[tex]\[ \text{Total price} = 1 \times s + 2 \times t \][/tex]
Substituting the values of [tex]\( s \)[/tex] and [tex]\( t \)[/tex]:
[tex]\[ \text{Total price} = 1 \times \frac{39}{2} + 2 \times \frac{17}{2} \][/tex]
This simplifies to:
[tex]\[ \text{Total price} = \frac{39}{2} + \frac{34}{2} = \frac{73}{2} = 36.50 \][/tex]
So, the total price of Linda's purchase is:
[tex]\[ \$ 36.50 \][/tex]
Therefore, the total price, in dollars and cents, of Linda's purchase is \$36.50.
