Question

1. $\

$$\begin{cases} x+5y=15 \\\\ x+3y=2 \\end{cases}$$

$

1. $\

$$\begin{cases} 2x+y=11 \\\\ -2x-3y=-13 \\end{cases}$$

$

1. $\

$$\begin{cases} 3x-8y=-11 \\\\ 3x-2y=-5 \\end{cases}$$

$

1. $\

$$\begin{cases} x-6y=-22 \\\\ -10x+12y=28 \\end{cases}$$

$

1. $\

$$\begin{cases} -20x-10y=30 \\\\ -10x+7y=3 \\end{cases}$$

$

1. $\

$$\begin{cases} -10x-6y=5 \\\\ -5x-3y=-2 \\end{cases}$$

$

1. $\

$$\begin{cases} y-3x=11 \\\\ y+7x=11 \\end{cases}$$

$

1. $\

$$\begin{cases} y+4x=1 \\\\ y-2x=-5 \\end{cases}$$

$

Explanation:

Step1: Solve first system via elimination

Subtract the second equation from the first:
$$(x+5y)-(x+3y)=15-2$$
$$2y=13 \implies y=\frac{13}{2}=6.5$$
Substitute $y=6.5$ into $x+3y=2$:
$$x+3(6.5)=2 \implies x=2-19.5=-17.5$$

Step2: Solve second system via elimination

Add the two equations:
$$(2x+y)+(-2x-3y)=11+(-13)$$
$$-2y=-2 \implies y=1$$
Substitute $y=1$ into $2x+y=11$:
$$2x+1=11 \implies 2x=10 \implies x=5$$

Step3: Solve third system via elimination

Subtract the second equation from the first:
$$(3x-8y)-(3x-2y)=-11-(-5)$$
$$-6y=-6 \implies y=1$$
Substitute $y=1$ into $3x-2y=-5$:
$$3x-2(1)=-5 \implies 3x=-3 \implies x=-1$$

Step4: Solve fourth system via scaling

Multiply first equation by 5: $5x-30y=-110$
Multiply second equation by 1: $-10x+12y=28$
Multiply first new equation by 2: $10x-60y=-220$
Add to second equation:
$$(10x-60y)+(-10x+12y)=-220+28$$
$$-48y=-192 \implies y=4$$
Substitute $y=4$ into $x-6y=-22$:
$$x-6(4)=-22 \implies x=24-22=2$$

Step5: Solve fifth system via scaling

Divide first equation by -10: $2x+y=-3$
Multiply this by 5: $10x+5y=-15$
Add to second equation $-10x+7y=3$:
$$(10x+5y)+(-10x+7y)=-15+3$$
$$12y=-12 \implies y=-1$$
Substitute $y=-1$ into $2x+y=-3$:
$$2x-1=-3 \implies 2x=-2 \implies x=-1$$

Step6: Solve sixth system via scaling

Multiply second equation by -2: $10x+6y=4$
Subtract first equation $-10x-6y=5$:
$$(10x+6y)-(-10x-6y)=4-5$$
$$20x+12y=-1$$
Substitute $10x=-6y+5$ (from first equation):
$$2(-6y+5)+12y=-1 \implies -12y+10+12y=-1 \implies 10=-1$$
This is a contradiction, so no solution.

Step7: Solve seventh system via elimination

Subtract the first equation from the second:
$$(y+7x)-(y-3x)=11-11$$
$$10x=0 \implies x=0$$
Substitute $x=0$ into $y-3x=11$:
$$y-0=11 \implies y=11$$

Step8: Solve eighth system via elimination

Subtract the second equation from the first:
$$(y+4x)-(y-2x)=1-(-5)$$
$$6x=6 \implies x=1$$
Substitute $x=1$ into $y+4x=1$:
$$y+4(1)=1 \implies y=1-4=-3$$

Answer:

1. $x=-17.5, y=6.5$
2. $x=5, y=1$
3. $x=-1, y=1$
4. $x=2, y=4$
5. $x=-1, y=-1$
6. No solution
7. $x=0, y=11$
8. $x=1, y=-3$