To determine the cost of a birthday card [tex]\( x \)[/tex] and the cost of a thank-you note [tex]\( y \)[/tex], we need to solve the system of linear equations given by:
[tex]\[
\left\{
\begin{array}{l}
3x + 2y = 9.6 \\
8x + 6y = 26.6
\end{array}
\right.
\][/tex]
Here's a detailed, step-by-step procedure to solve the system:
1. Write the two equations in standard form:
- Equation 1: [tex]\( 3x + 2y = 9.6 \)[/tex]
- Equation 2: [tex]\( 8x + 6y = 26.6 \)[/tex]
2. Express the system of equations in matrix form:
[tex]\[
\begin{pmatrix}
3 & 2 \\
8 & 6
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
9.6 \\
26.6
\end{pmatrix}
\][/tex]
3. Determine the inverse of the coefficient matrix (if possible) and solve the matrix equation. But for simplicity, we usually proceed with methods such as Gaussian elimination or substitution in hand calculations.
4. Solve the equations using elimination method:
- Multiply the first equation by 3 to make the coefficients of [tex]\( y \)[/tex] same:
[tex]\[
3(3x + 2y) = 3(9.6) \\
9x + 6y = 28.8
\][/tex]
- Now we have:
[tex]\[
9x + 6y = 28.8 \qquad \text{(Equation 3)} \\
8x + 6y = 26.6 \qquad \text{(Equation 2)}
\][/tex]
- Subtract Equation 2 from Equation 3:
[tex]\[
(9x + 6y) - (8x + 6y) = 28.8 - 26.6 \\
x = 2.2
\][/tex]
- So, the cost of one birthday card [tex]\( x \)[/tex] is \[tex]$2.20.
5. Substitute \( x = 2.2 \) back into one of the original equations to find \( y \):
- Use the first equation:
\[
3(2.2) + 2y = 9.6 \\
6.6 + 2y = 9.6 \\
2y = 9.6 - 6.6 \\
2y = 3 \\
y = 1.5
\]
So, from this step-by-step procedure, we conclude:
- The cost of a birthday card \( x \) is \(\$[/tex]2.20\).
- The cost of a thank-you note [tex]\( y \)[/tex] is [tex]\(\$1.50\)[/tex].
Therefore, the accurate cost of a birthday card is:
[tex]\(\boxed{2.20}\)[/tex].
