Question

1. (1,5) ;

-7m + 8n = 33

1. (-2,0) ; 50 = -9x - 2y

(7, -4) ;
9b + 4a = -8
6a + 5b - 42 = 0

1. (-3,-2) ;

-7c - 4d = 29
3c = -7 + d
b) 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
c) write a system of linear equations that has the solution (4, 3).

Explanation:

Part 1: Verify given points as solutions

Step1: Check (1,5) in eq1

Substitute $m=1, n=5$ into $-7m+8n=33$:

$$\begin{align*} -7(1)+8(5)&=-7+40=33 \end{align*}$$

Step2: Confirm (1,5) is solution

Both sides equal, so it is a solution.

Step3: Check (-2,0) in eq2

Substitute $x=-2, y=0$ into $50=-9x-2y$:

$$\begin{align*} -9(-2)-2(0)&=18+0=18 eq50 \end{align*}$$

Step4: Confirm (-2,0) not solution

Sides not equal, so not a solution.

Step5: Check (7,-4) in eq3

Substitute $a=7, b=-4$ into $9b+4a=-8$:

$$\begin{align*} 9(-4)+4(7)&=-36+28=-8 \end{align*}$$

Substitute into $6a+5b-42=0$:

$$\begin{align*} 6(7)+5(-4)-42&=42-20-42=-20 eq0 \end{align*}$$

Step6: Confirm (7,-4) not solution

Fails second equation, so no.

Step7: Check (-3,-2) in eq4

Substitute $c=-3, d=-2$ into $-7c-4d=29$:

$$\begin{align*} -7(-3)-4(-2)&=21+8=29 \end{align*}$$

Substitute into $3c=-7+d$:

$$\begin{align*} 3(-3)&=-9;\ -7+(-2)=-9 \end{align*}$$

Step8: Confirm (-3,-2) is solution

Both equations hold, so yes.

Step1: Check (6,9) in system 5

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

$$\begin{align*} 6+7(9)&=6+63=69 \end{align*}$$

Substitute into $6t+4s=78$:

$$\begin{align*} 6(9)+4(6)&=54+24=78 \end{align*}$$

Step2: Confirm (6,9) for system5

Both equations hold, so yes.

Step3: Check (6,9) in system6

Substitute $p=6, q=9$ into $-2p+5q=34$:

$$\begin{align*} -2(6)+5(9)&=-12+45=33 eq34 \end{align*}$$

Step4: Confirm (6,9) for system6

Fails first equation, so no.

Step1: Create first linear equation

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

Step2: Create second linear equation

Use $x-y=k$, substitute $x=4,y=3$: $4-3=1$, so $x-y=1$.

Answer:

1. yes
2. no
3. no
4. yes

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Part B: Check if (6,9) is a solution