Answer :
To solve the problem, we need to set up equations based on the conditions given and identify which of the provided augmented matrices correspond to those equations.
### Step 1: Understand the Problem
Liam wants to pay for his car in three installments: [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex], where:
- [tex]\(x\)[/tex] is the first installment.
- [tex]\(y\)[/tex] is the second installment.
- [tex]\(z\)[/tex] is the third installment.
We are given the following conditions:
1. The total cost of the car is [tex]\(\$29,000\)[/tex].
2. Two times the first installment ([tex]\(2x\)[/tex]) is [tex]\(\$1,000\)[/tex] more than the sum of the third installment ([tex]\(z\)[/tex]) and three times the second installment ([tex]\(3y\)[/tex]).
3. A [tex]\(15\%\)[/tex] interest is charged on the second and third installments, amounting to [tex]\(\$2,100\)[/tex].
### Step 2: Set Up the Equations
From the conditions, we can write the following equations:
1. Total Cost Equation:
[tex]\[
x + y + z = 29,000
\][/tex]
2. First Installment Equation:
[tex]\[
2x = 3y + z + 1,000
\][/tex]
Rearrange it to:
[tex]\[
2x - 3y - z = 1,000
\][/tex]
3. Interest Equation:
[tex]\[
0.15y + 0.15z = 2,100
\][/tex]
This can also be expressed as:
[tex]\[
0 \cdot x + 0.15y + 0.15z = 2,100
\][/tex]
### Step 3: Write the Augmented Matrix
Each equation corresponds to a row in the augmented matrix, resulting in:
[tex]\[
\begin{bmatrix}
1 & 1 & 1 & \vert & 29,000 \\
2 & -3 & -1 & \vert & 1,000 \\
0 & 0.15 & 0.15 & \vert & 2,100
\end{bmatrix}
\][/tex]
### Step 4: Match the Augmented Matrix to Given Options
Compare our derived augmented matrix with the given matrices. The correct matrices are:
- The matrix matches the setup we derived:
[tex]\[
\begin{bmatrix}
1 & 1 & 1 & \vert & 29,000 \\
2 & -3 & -1 & \vert & 1,000 \\
0 & 0.15 & 0.15 & \vert & 2,100
\end{bmatrix}
\][/tex]
This corresponds to the first and third matrices in the provided options.
So, the indices of the correct matrices are [0, 2].
### Step 1: Understand the Problem
Liam wants to pay for his car in three installments: [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex], where:
- [tex]\(x\)[/tex] is the first installment.
- [tex]\(y\)[/tex] is the second installment.
- [tex]\(z\)[/tex] is the third installment.
We are given the following conditions:
1. The total cost of the car is [tex]\(\$29,000\)[/tex].
2. Two times the first installment ([tex]\(2x\)[/tex]) is [tex]\(\$1,000\)[/tex] more than the sum of the third installment ([tex]\(z\)[/tex]) and three times the second installment ([tex]\(3y\)[/tex]).
3. A [tex]\(15\%\)[/tex] interest is charged on the second and third installments, amounting to [tex]\(\$2,100\)[/tex].
### Step 2: Set Up the Equations
From the conditions, we can write the following equations:
1. Total Cost Equation:
[tex]\[
x + y + z = 29,000
\][/tex]
2. First Installment Equation:
[tex]\[
2x = 3y + z + 1,000
\][/tex]
Rearrange it to:
[tex]\[
2x - 3y - z = 1,000
\][/tex]
3. Interest Equation:
[tex]\[
0.15y + 0.15z = 2,100
\][/tex]
This can also be expressed as:
[tex]\[
0 \cdot x + 0.15y + 0.15z = 2,100
\][/tex]
### Step 3: Write the Augmented Matrix
Each equation corresponds to a row in the augmented matrix, resulting in:
[tex]\[
\begin{bmatrix}
1 & 1 & 1 & \vert & 29,000 \\
2 & -3 & -1 & \vert & 1,000 \\
0 & 0.15 & 0.15 & \vert & 2,100
\end{bmatrix}
\][/tex]
### Step 4: Match the Augmented Matrix to Given Options
Compare our derived augmented matrix with the given matrices. The correct matrices are:
- The matrix matches the setup we derived:
[tex]\[
\begin{bmatrix}
1 & 1 & 1 & \vert & 29,000 \\
2 & -3 & -1 & \vert & 1,000 \\
0 & 0.15 & 0.15 & \vert & 2,100
\end{bmatrix}
\][/tex]
This corresponds to the first and third matrices in the provided options.
So, the indices of the correct matrices are [0, 2].
