Question

1. \\(\

$$\begin{cases}2x + 7y = 5 \\\\ 2x + 3y = 9\\end{cases}$$

\\)\

1. \\(\

$$\begin{cases}x + 6y = 48 \\\\ -x + y = 8\\end{cases}$$

\\)\

1. \\(\

$$\begin{cases}8x - 6y = -122 \\\\ -4x + 6y = 94\\end{cases}$$

\\)

Explanation:

For system 4:

Step1: Subtract equations

Subtract the second equation from the first to eliminate $x$.
$$(2x+7y)-(2x+3y)=5-9$$
$$4y=-4$$

Step2: Solve for $y$

Divide both sides by 4 to isolate $y$.
$$y=\frac{-4}{4}=-1$$

Step3: Substitute $y=-1$ into second equation

Plug $y=-1$ into $2x+3y=9$ to solve for $x$.
$$2x+3(-1)=9$$
$$2x-3=9$$
$$2x=12$$
$$x=6$$

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For system 5:

Step1: Add the two equations

Add the equations to eliminate $x$.
$$(x+6y)+(-x+y)=48+8$$
$$7y=56$$

Step2: Solve for $y$

Divide both sides by 7 to isolate $y$.
$$y=\frac{56}{7}=8$$

Step3: Substitute $y=8$ into second equation

Plug $y=8$ into $-x+y=8$ to solve for $x$.
$$-x+8=8$$
$$-x=0$$
$$x=0$$

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For system 6:

Step1: Add the two equations

Add the equations to eliminate $y$.
$$(8x-6y)+(-4x+6y)=-122+94$$
$$4x=-28$$

Step2: Solve for $x$

Divide both sides by 4 to isolate $x$.
$$x=\frac{-28}{4}=-7$$

Step3: Substitute $x=-7$ into second equation

Plug $x=-7$ into $-4x+6y=94$ to solve for $y$.
$$-4(-7)+6y=94$$
$$28+6y=94$$
$$6y=66$$
$$y=11$$

Answer:

1. $x=6$, $y=-1$
2. $x=0$, $y=8$
3. $x=-7$, $y=11$