Question

1. $4r + 6s - t = -18$

$3r + 2s - 4t = -24$
$-5r + 3s + 2t = 15$

Explanation:

Step1: Label the equations

Let \(

$$\begin{cases} 4r + 6s - t = -18 \quad (1)\\ 3r + 2s - 4t = -24 \quad (2)\\ -5r + 3s + 2t = 15 \quad (3) \end{cases}$$

\)

Step2: Eliminate \(t\) from (1) and (2), (1) and (3)

From (1): \(t = 4r + 6s + 18\) (4)
Substitute (4) into (2):
\(3r + 2s - 4(4r + 6s + 18) = -24\)
\(3r + 2s - 16r - 24s - 72 = -24\)
\(-13r - 22s = 48\) (5)

Substitute (4) into (3):
\(-5r + 3s + 2(4r + 6s + 18) = 15\)
\(-5r + 3s + 8r + 12s + 36 = 15\)
\(3r + 15s = -21\)
Simplify: \(r + 5s = -7\) (6)

Step3: Solve (5) and (6)

From (6): \(r = -7 - 5s\) (7)
Substitute (7) into (5):
\(-13(-7 - 5s) - 22s = 48\)
\(91 + 65s - 22s = 48\)
\(43s = -43\)
\(s = -1\)

Step4: Find \(r\)

Substitute \(s = -1\) into (7):
\(r = -7 - 5(-1) = -7 + 5 = -2\)

Step5: Find \(t\)

Substitute \(r = -2\), \(s = -1\) into (4):
\(t = 4(-2) + 6(-1) + 18 = -8 - 6 + 18 = 4\)

Answer:

\(r = -2\), \(s = -1\), \(t = 4\)