$\displaystyle 4x+3y=6$

$\displaystyle 2x+2y=4$

For the second equation, solve for $\displaystyle x$ in terms of $\displaystyle y$.

$\displaystyle 4-2x=2y$

$\displaystyle y=2-x$

Plug this value of y into the first equation.

$\displaystyle 4x+3(2-x)=6$

$\displaystyle 4x + 6- 3x =6$

$\displaystyle x=0$

$\displaystyle x+y=32$ 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 $\displaystyle (2, 10)$

To find the point where these two lines intersect, set the equations equal to each other, such that $\displaystyle y$ is substituted with the $\displaystyle x$ side of the second equation. Solving this new equation for $\displaystyle x$ will give the $\displaystyle x$-coordinate of the point of intersection.

$\displaystyle 3x = x - 2$

Subtract $\displaystyle x$ from both sides.

$\displaystyle (3x) - x = (x - 2) - x$

$\displaystyle 2x = - 2$

Divide both sides by 2.

$\displaystyle \frac{2x}{2} = \frac{-2}{2}$

$\displaystyle x = - 1$

Now substitute $\displaystyle -1$ into either equation to find the $\displaystyle y$-coordinate of the point of intersection.

$\displaystyle y = 3x$

$\displaystyle y= 3(-1)$

$\displaystyle y = -3$

With both coordinates, we know the point of intersection is $\displaystyle (-1,-3)$. One can plug in $\displaystyle -1$ for $\displaystyle x$ and $\displaystyle -3$ for $\displaystyle y$ in both equations to verify that this is correct.

$\displaystyle 3x - 5y = 5$

$\displaystyle -2x + 5y = 0$

Add the equations together.

$\displaystyle 3x+(-2x)=x$

$\displaystyle -5y+5y=0$

$\displaystyle 5+0=5$

Put the terms together to see that $\displaystyle x+0=5\ \textup{or}\ x=5$.

Substitute this value into one of the original equaitons and solve for $\displaystyle y$.

$\displaystyle 3x - 5y = 5$

$\displaystyle 3(5) - 5y = 5$

$\displaystyle -5y=-10$

$\displaystyle y=2$

Now we know that $\displaystyle x=5\ \textup{and}\ y=2$, thus we can find the sum of $\displaystyle x$ and $\displaystyle y$.

$\displaystyle x+y=5+2=7$

We can solve this problem by setting up a simple system of equations. First, we want to change the equations so one variable can cancel out. Multiplying the first equation by 2 and the second equation by 3 gives us a new system of $\displaystyle 2y = 6x + 8$ and $\displaystyle 3y=-6x+27$. These equations add up to $\displaystyle 5y=35$ or $\displaystyle y = 7$. Plugging in 7 for $\displaystyle y$ in either of the original two equations shows us that $\displaystyle x$ is equal to 1 and the point is $\displaystyle (1,7)$.

This system has infinite solutions becasue the two equations are actually the exact same line. To discover this, put both equations in terms of y.

First, $\displaystyle 2x - y = -3$. Add y to both sides:

$\displaystyle 2x = -3 + y$ Now add 3 to both sides:

$\displaystyle 2x + 3 = y$

Now we can show that the second equation also represents the line $\displaystyle y=2x + 3$

$\displaystyle 2y-6 = 4x$ add 6 to both sides

$\displaystyle 2y = 4x + 6$ divide both sides by 2

$\displaystyle y=2x+3$

Since both equations are the same line, literally any point on one line will also be on the other - infinite solutions.

A system of equations can be solved by subsituting one variable for another. Since we know that $\displaystyle y=x-2$, we can subsitute this into the other equation so $\displaystyle x-2=2x-4$. This expression can be solved to find that $\displaystyle x=-6$. Now that we know the value of $\displaystyle x$ it can be subsituted into either of the original equations to find $\displaystyle y$.

$\displaystyle \\\textup{Pig, Dog and Cat are } P, D, \textup{ and } C \textup{ years old, respectively.}\\\textup{The three statements translate in the following three equations, respectively}\\P-2=3(D-2)\\P+1=3(C+1)\\P+2=2.5(C+2).\\\textup{ Subtract the first from the second equation and obtain }\\(P-2)-(P+1)=(3D-6)-(3C+3)\\-3=3(D-C)-9\\D-C=2.\\\textup{So Dog is two years older than Cat.}$

To find the intersection point, you must find the values of x and y that satisfy the two equations. We can use the method of adding the two equations together:

$\displaystyle 3x+ 2y=4$

$\displaystyle 5x-2y=4$

If we add the equations together, the y terms cancel out, so we get

$\displaystyle 8x=8$

$\displaystyle x=1$

Now that we know the value of x, we can plug that into one of the equations and solve for y. Pluggin it into the first equation, we get

$\displaystyle 3(1)+2y=4$

$\displaystyle 2y=1$

$\displaystyle y=\frac{1}{2}$.

So that point of intersection is $\displaystyle (x,y)$, or $\displaystyle \left(1,\frac{1}{2}\right)$.