Process all the system of equations
by graphing.
When we graph two lines on the coordinate plane,
three things can happen.
Number one, there could be one point of intersection,
and therefore, there's one solution.
Number two, lines could be parallel and never intersect,
and therefore, there are no solutions.
Or we can graph the two lines,
and the two lines are identical or coincide.
In that case there's an infinite number of solutions.
So let's begin by graphing two x plus y equals four.
Let's go and write this equation in slope-intercept form,
which is the form y equals mx plus b,
and then we'll graph the line using the y-intercept
or vertical intercept, as well as the slope.
So beginning with two x plus y equals four.
To solve for y, we subtract two x on both sides.
Simplifying.
Simplifying on the left, two x minus two x is zero,
which is why we subtracted two x on both sides.
So now on the left, we just have y equals.
On the right, we have four minus two x,
which I'm going to write as negative two x plus four,
so the equation is in slope-intercept form.
So now we know the slope m is equal to negative two,
the coefficient of x,
which we'll write as negative two over one.
Whenever using slope to graph,
we do want it to be in fraction form or a ratio
because the slope is equal to the change of y
over the change of x.
And then b is equal to positive four,
and therefore, the vertical or y-intercept
is the ordered pair zero comma four.
So now we plot the vertical intercept
at zero comma four which is here.
And then from here,
we use a slope to find additional points.
Again, because the slope is negative two over one
where negative two is the change of y,
sometimes referred to as the rise,
and positive one is a change of x,
sometimes referred to as the run,
from the vertical intercept we go down two units
and write one.
And we can do this over and over again.
Down two, write one. Down two, write one.
Down two, write one, and so on.
Now if we wanted to go up instead of down,
if the slope is equal to negative two over positive one,
that's equivalent to positive two over negative one,
which means if we go back to the vertical intercept,
because the slope is positive two over negative one,
we would go up two and left one,
up two and left one, and so on.
Let's go and sketch the line.
And now let's graph the second line.
Let's go and do this in black,
where x minus y is equal to two.
Now we could go ahead and solve this for y
but since the left side is x minus y,
it'll be easier just to make a table of values
to graph this line.
So go ahead and make a table of values.
We just need to pick x and y,
so that x minus y is equal to two.
For example, if x is equal to two,
y would be equal to zero since two minus zero is two.
If the x is three, since three minus one is two, y is one.
If x is four, y would have to be two
since four minus two is two.
Let's do this one more time.
When x is five, since five minus three is two,
y has to be three.
Let's go and plot these points,
two comma zero, three comma one,
four comma two, five comma three.
But looking at these points,
we can see the slope or the change of y over the change of x
is going to be up one, right one.
So if we went to, we can use that slope
to find additional points.
Up one, right one. Up one, right one.
Up one, right one which is equivalent
to going down one and back one.
Down one, back one, and so on.
Let's go and sketch the line.
Notice how these two lines do intersect.
The point of intersection represents the solution.
So the ordered pair at this point of intersection
gives us the solution.
If you can see is two comma zero which means the solution,
and an ordered pair is two comma zero,
or we can also say x equals two, y equals zero.
And before we go,
let's verify these values do satisfy both equations.
To do this for both equations we substitute two for x
and zero for y.
Performing the substitution into the first equation
gives us two times x which is two,
plus y which is zero, equals four.
Simplifying we do have four equals four, which is true.
Verifying, the ordered pair does satisfy the first equation.
And now for the second equation.
Again, substituting two for x and zero for y
gives us two minus zero equals two, which is true.
Two equals two.
And therefore, once again,
the ordered pair does satisfy the second equation,
which verifies our solution is correct.
x equals two. y equals zero.
We have one solution.
I hope you found this helpful.