Answer :
We begin by defining the variables as follows:
- Let [tex]\( x \)[/tex] be the first installment.
- Let [tex]\( y \)[/tex] be the second installment.
- Let [tex]\( z \)[/tex] be the third installment.
The problem gives us three conditions:
1. The total cost of the car is \[tex]$29,000. This yields:
$[/tex][tex]$
x + y + z = 29000.
$[/tex][tex]$
2. Two times the first installment is \$[/tex]1,000 more than the sum of the third installment and three times the second installment. In equation form, we have:
[tex]$$
2x = z + 3y + 1000.
$$[/tex]
Rearranging the equation by subtracting [tex]\( 3y \)[/tex] and [tex]\( z \)[/tex] from both sides, we obtain:
[tex]$$
2x - 3y - z = 1000.
$$[/tex]
3. Liam must pay 15% interest on the second and the third installments, and this interest amounts to \[tex]$2,100. The interest on \( y \) and \( z \) is given by:
$[/tex][tex]$
0.15y + 0.15z = 2100.
$[/tex][tex]$
The augmented matrix representing these three equations is constructed by writing the coefficients of \( x \), \( y \), and \( z \) in each equation along with the constant term. Thus, the augmented matrix is:
$[/tex][tex]$
\left[\begin{array}{ccc|r}
1 & 1 & 1 & 29000 \\
2 & -3 & -1 & 1000 \\
0 & 0.15 & 0.15 & 2100
\end{array}\right].
$[/tex]$
This is the correct augmented matrix that models Liam's situation.
- Let [tex]\( x \)[/tex] be the first installment.
- Let [tex]\( y \)[/tex] be the second installment.
- Let [tex]\( z \)[/tex] be the third installment.
The problem gives us three conditions:
1. The total cost of the car is \[tex]$29,000. This yields:
$[/tex][tex]$
x + y + z = 29000.
$[/tex][tex]$
2. Two times the first installment is \$[/tex]1,000 more than the sum of the third installment and three times the second installment. In equation form, we have:
[tex]$$
2x = z + 3y + 1000.
$$[/tex]
Rearranging the equation by subtracting [tex]\( 3y \)[/tex] and [tex]\( z \)[/tex] from both sides, we obtain:
[tex]$$
2x - 3y - z = 1000.
$$[/tex]
3. Liam must pay 15% interest on the second and the third installments, and this interest amounts to \[tex]$2,100. The interest on \( y \) and \( z \) is given by:
$[/tex][tex]$
0.15y + 0.15z = 2100.
$[/tex][tex]$
The augmented matrix representing these three equations is constructed by writing the coefficients of \( x \), \( y \), and \( z \) in each equation along with the constant term. Thus, the augmented matrix is:
$[/tex][tex]$
\left[\begin{array}{ccc|r}
1 & 1 & 1 & 29000 \\
2 & -3 & -1 & 1000 \\
0 & 0.15 & 0.15 & 2100
\end{array}\right].
$[/tex]$
This is the correct augmented matrix that models Liam's situation.
