Isolate $\displaystyle y$ in the first equation.

$\displaystyle y=16-4x$

Plug $\displaystyle y$ into the second equation to solve for $\displaystyle x$.

$\displaystyle 2x+3(16-4x)=18$

$\displaystyle 2x+48-12x=18$

$\displaystyle -10x+48=30$

$\displaystyle -10x=-30$

$\displaystyle x=3$

Plug $\displaystyle x$ into the first equation to solve for $\displaystyle y$.

$\displaystyle 4(3)+y=16$

$\displaystyle 12+y=16$

$\displaystyle y=4$

Now we have both the $\displaystyle x$ and $\displaystyle y$ values and can express them as a point: $\displaystyle (3,4)$.

1st equation: $\displaystyle 3x+2y=2$

2nd equation: $\displaystyle 2x+2y=4$

Subtract the 2nd equation from the 1st equation to eliminate the "2y" from both equations and get an answer for x:

$\displaystyle x=-2$

Plug the value of $\displaystyle x$ into either equation and solve for $\displaystyle y$:

$\displaystyle 2(-2)+2y=4$

$\displaystyle -4+2y=4$

$\displaystyle 2y=4+4$

$\displaystyle 2y=8$

$\displaystyle y=4$

1. $\displaystyle 2y=3x+11$
2. $\displaystyle y=-5x-14$

Substitute equation 2. into equation 1.,

$\displaystyle 2\left(-5x-14\right)=3x+11$

$\displaystyle -10x-28=3x+11$

$\displaystyle -13x=39$

so, $\displaystyle x=-3$

Substitute $\displaystyle x=-3$ into equation 2:

$\displaystyle y=-5\cdot (-3)-14=1$

so, the solution is $\displaystyle (-3,1)$.

This problem is a system of equations, and uses the equation $\displaystyle \text{distance}=\text{rate}\times\text{time}$.  

Start by assigning variables. Let $\displaystyle b$ stand for the rate of the boat, let $\displaystyle c$ stand for the rate of the current.  

When the boat is going upstream, the total rate is equal to $\displaystyle b-c$.  You must subtract because the rates are working against each other—the boat is going slower than it would because it has to work against the current.

Using our upstream distance (400m) and time (2hr) from the question, we can set up our rate equation:

$\displaystyle 400 = (b-c) \times 2$

When the boat is going downstream, the total rate is equal to $\displaystyle b+c$ because the boat and current are working with each other, causing the boat to travel faster.

We can refer to the downstream distance (600m) and time (2hr) to set up the second equation:

 $\displaystyle 600 = (b+c) \cdot 2$

From here, use elimination to solve for $\displaystyle b$ and $\displaystyle c$.

1. Set up the system of equations, and solve for $\displaystyle b$.

$\displaystyle 400 = 2b - 2c$ 
$\displaystyle 600 = 2b + 2c$

$\displaystyle 1000 = 4b$

$\displaystyle 250 = b$

2. Subsitute $\displaystyle b$ into one of the equations to solve for $\displaystyle c$.

$\displaystyle 600 = 2(250) + 2c$   
$\displaystyle 600 = 500 + 2c$

$\displaystyle 100 = 2c$

$\displaystyle c= 50$

Step 1: Set up the equations

Let $\displaystyle n$= Nick's age now
Let $\displaystyle s$= Sarah's age now

The first part of the question says "Nick's sister is three times as old as him".  This means:

$\displaystyle s=3n$

The second part of the equation says "in two years, she will be twice as old as he is then).  This means:

$\displaystyle s+2=2(n+2)$

Add 2 to each of the variables because each of them will be two years older than they are now.

Step 2: Solve the system of equations using substitution

Substitute $\displaystyle 3n$ for $\displaystyle s$ in the second equation.  Solve for $\displaystyle n$

$\displaystyle 3n+2=2(n+2)$

$\displaystyle 3n+2=4n+4$

$\displaystyle n+2=4$

$\displaystyle {\color{Red} n=2}$

Plug $\displaystyle n$ into the first equation to find $\displaystyle s$

$\displaystyle s=3(2)$

$\displaystyle {\color{Red} s=6}$

Solve using elimination:

$\displaystyle 4x-6y =12$

$\displaystyle 2(2x+2y=6)$ multiply the 2nd equation by two to make elimination possible
________________
$\displaystyle 4x-6y=12$
$\displaystyle 4x+4y=12$ subtract 2nd equation from the first to solve for $\displaystyle y$
________________
$\displaystyle -10y=0$
$\displaystyle {\color{Red} y=0}$

Substitute $\displaystyle y$ into either equation to solve for $\displaystyle x$

$\displaystyle 2x+2(0)=6$
$\displaystyle 2x=6$
$\displaystyle {\color{Red} x=3}$

There are two ways to solve this:

-The 1st equation can be mutliplied by $\displaystyle 2$ while the 2nd equation can be multiplied by $\displaystyle -3$ and added to the 1st equation to make it a single variable equation where

$\displaystyle 22y=-66$  

$\displaystyle y=-3$.

This can be plugged into either equation to get $\displaystyle x=2$

or

-The 2nd equation can be simplified to,

$\displaystyle x-2y=8$  

$\displaystyle x=8+2y$.

This value for $\displaystyle x$ can then be substituted into the first equation to make the equation single variable in $\displaystyle y, 3(8+2y)+5y=-9$.

Solving, gives $\displaystyle y=-3$, which can be plugged into either original equation to get $\displaystyle x=2$

$\displaystyle x+y=32$ rearranges to

$\displaystyle x=32-y$

and

$\displaystyle x=\frac{2}{3}y-2$, so

$\displaystyle 32-y=\frac{2}{3}y-2$

$\displaystyle 32-1\frac{2}{3}y=-2$

$\displaystyle -1\frac{2}{3}y=-34$

$\displaystyle -\frac{5}{3}y=-34$

$\displaystyle y=\frac{-34}{1}\cdot\frac{-3}{5}=\frac{102}{5}=20\frac{2}{5}$

$\displaystyle x=32-20\frac{2}{5}=11\frac{3}{5}$

In the second equation, you can substitute $\displaystyle 3x + 4$ for $\displaystyle y$ from the first.

$\displaystyle 2x + 3y = 34$

$\displaystyle 2x + 3 (3x + 4) = 34$

$\displaystyle 2x + 3 (3x) + 3 (4) = 34$

$\displaystyle 2x + 9x + 12 = 34$

$\displaystyle 11x + 12 = 34$

$\displaystyle 11x = 22$

$\displaystyle x = 2$

Now, substitute 2 for $\displaystyle x$ in the first equation:

$\displaystyle y = 3x + 4$

$\displaystyle y = 3 (2) + 4$

$\displaystyle y = 6 + 4$ 

$\displaystyle y = 10$

The solution is $\displaystyle (2, 10)$

Use elimination, multiply the top equation by -4 so that you can eliminate the X's.

$\displaystyle x-3y=-22$

$\displaystyle 4x+2y=-4$

 __________________

$\displaystyle -4x+12y=88$

$\displaystyle 4x+2y=-4$

Combine these two equations, and then you have;

$\displaystyle 14y=84$

$\displaystyle y=6$

Plug in y into one of the original equations and solve for x.

$\displaystyle x-3(6)=-22$

$\displaystyle x-18=-22$

$\displaystyle x=-4$

Your solution is $\displaystyle (-4,6)$.