Question

check whether (6, 9) is a solution of the systems of linear equations.

1. ( s + 7t = 69 )

( 6t + 4s = 78 )

1. ( -2p + 5q = 34 )

( -7q = -61 - 8p )
write a system of linear equations that has the solution (4, 3).

Explanation:

For System 5: $(s,t)=(6,9)$

Step1: Substitute into first equation

Substitute $s=6$, $t=9$ into $s+7t=69$:
$6 + 7\times9 = 6 + 63 = 69$

Step2: Substitute into second equation

Substitute $s=6$, $t=9$ into $6t+4s=78$:
$6\times9 + 4\times6 = 54 + 24 = 78$

For System 6: $(p,q)=(6,9)$

Step1: Substitute into first equation

Substitute $p=6$, $q=9$ into $-2p+5q=34$:
$-2\times6 + 5\times9 = -12 + 45 = 33
eq 34$

Step2: Verify second equation (optional)

Substitute $p=6$, $q=9$ into $-7q=-61-8p$:
$-7\times9 = -63$; $-61 -8\times6 = -61-48=-109$, $-63
eq-109$

For creating system with $(4,3)$

Step1: Make first linear equation

Use $x+y$: $4+3=7$, so $x+y=7$

Step2: Make second linear equation

Use $2x-y$: $2\times4 -3=5$, so $2x-y=5$

Answer:

1. For the system

$$\begin{cases}s + 7t = 69\\6t + 4s = 78\end{cases}$$

, $(6,9)$ is a solution.

1. For the system

$$\begin{cases}-2p + 5q = 34\\-7q = -61 - 8p\end{cases}$$

, $(6,9)$ is not a solution.

1. A sample system with solution $(4,3)$:

$$\begin{cases}x + y = 7\\2x - y = 5\end{cases}$$

(other valid systems are also acceptable)