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:

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

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

.

So that point of intersection is , or .

There are several methods to find the solution set to this system of equations, but here we will use the substitution method for its ease of application. The idea of this method is to solve one of the equations (and you can choose either) in terms of one variable and then plug it into the second equation that was not tampered with to solve it for the other variable. Do not worry if this isn't completely clear just yet. But after you read the step-by-step solution review this paragraph and make sure the math language and this explanation align in your mind.

1) 

2) 

First solve equation 2 for y. Again, it is your choice which equation and which variable to use, but try and select one that will take the least work. In our case neither are "better", but the bottom equation will not yield a fraction.

move over x term

isolate y

Now we have equation 2 in terms of one variable (y=....). We must now plug this equation we just made back into one of the original equations to solve that equation for one variable. Let's do equation 1:

Notice how all we've done is replace the y in equation 1 with the modified version of equation 2. Now solve for x:

Simplify the parenthesis:

Combine x's and move the constant over:

Now we have solved for one variable and are almost done! Plug this value into either original equation and solve for y. Let's plug it into equation 2:

move the constant over after simplifying:

Final answer= 

We now have solved for each variable. Thus our solution set to this system of equations, where they are equal to each other, or where their lines intersect is (2.2,-0.4)! Since there are multiple ways to solve this problem, there are mutiple ways to check yourself. Convert both equations to slope intercept form, graph them using a graphing utility, and use the trace or intersect function to see that these two lines really do intersect and therefore equal each other at this coordinate point. Alternately just plug in the coordinate pair to either ORIGINAL equation, but the graphical method is probably easy since we have decimals.

The question is asking the student to solve the linear set of equations ultimately by isolating and .

There are a few ways a student could choose to answer this. 

One may see immediately  and realize to eliminate the result from the second equation one would subtract as follows.

--

Other students may choose to subtract the first equation twice from the second equation then subsequently solve for y.

The point these two equations intersect is their solution. There are several ways to solve and one should choose the method that makes the problem easiest. Here one can easily eliminate the two 4x variables. Thus, this problem will be solved by elimination.

1) 

2) 

Multiply equation 1 by -1:

Add equation 1 to equation 2:

1a)  

2)    

----------------------

Now substitute for y in either of the original equations:

2) 

The solution set to this system of equations is a coordinate point whose x and y values would satisfy the two equations. This is the same place the two equations intersect. The elimination method will be used to solve for the solution set.

1) 

2) 

Multiply equation 1 by -2:

Add the new equation 1 to equation 2:

--------------------------

Substitute this x value back into either original equation to solve for y:

First step is to solve one of the equations for one of the variables.

Choose the equation easiest to solve for one of the variables.

.             

Substitute  for  in the other equation and solve for .

Use distributive property.

Substitute  for  in either equation and solve for .

The solution is 

For elimination you need to get one variable by itself by cancelling the other out. In this equation this is best done by getting rid of . You can multiply whichever equation you would like to, but multiply it by  to get

then add the equations together

which, simplified, is

divied by  to get  

Then plug  back into any equation for the x value

Solve for  to get

The first step is to set the two equations equal to each other, since at the point of intersection, they will be. To do this more easily, convert each equation into slope-intercept form.

First equation:

 State equation

 Add  to both sides.

Second equation:

 State equation

 Divide both sides by .

 Subtract  from both sides.

Now, since each equation equals , the equations also equal each other (for the point of intersection). By solving for , therefore, we can 

 State equation.

 Subtract  from both sides.

 Subtract  from both sides.

 Divide both sides by .

So, the -coordinate of our intersection is . To find the -coordinate, plug this result back into one of the original equations.

 State your chosen equation.

 Substitute the value of .

So, the coordinates where the two lines intersect are .

This solution set is special, as one of our two lines is either horizontal or vertical. If the equation of a line in the coordinate plane contains only an  or a  variable, but not both, then the line is either horizontal (if only  is in the equation) or vertical (if only  is in the equation). This is good news, as solving for this intersection is much faster.

First, solve for  in our first equation.

 State the equation.

 Divide both sides by .

Now that we know , we simply substitute that value into our second equation.

 State the equation.

 Substitute the value of .

 Simplify.

 Subtract  (or ) from both sides.

 Divide both sides by  (or multiply both sides by ).

Thus, the intersection point of our two lines is at .

The first step is to set the two equations equal to each other, since at the point of intersection, they will be. To do this more easily, convert each equation into slope-intercept form.

First equation:

 Already in slope-intercept form

Second equation:

 State equation

 Add  to both sides.

 Divide both sides by .

At this point, note that both equations have idential slopes:  for both equations, but different -intercepts. Thus, the lines are parallel, and will never touch. We can stop here, but let's prove our theory with algebra by setting the equations equal to one another:

 Set your equations.

 Subtract  from both sides.

 No solution.

Thus, there is no solution to this equation, and the lines are parallel.