We're asked to solve the system of equations
by graphing.
If there is a solution,
the two lines would intersect in a point.
The ordered pair of the point of intersection
is the solution.
However, if the two lines are parallel,
there is no solution.
If the two lines coincide or are the same line,
there is an infinite number of solutions.
So our first equation is negative x plus y equals one.
Let's go ahead and write the equation
in slope-intercept form where the form y equals mx plus b,
where m is the slope, and b is the y or vertical intercept.
To solve for y, notice here,
we add x to both sides of the equation.
Simplifying.
We have y equals, let's write the right side as x plus one.
Or better yet, how about one x plus one.
Notice in this form, we can see the slope m,
which is the coefficient of x is equal to one,
but we always want the slope to be in a fraction form
so that we can use the slope to graph.
So let's write one as one over one.
And then b, the vertical intercept,
is equal to positive one,
which means the vertical or y-intercept
is the ordered pair zero comma one.
And remember the slope is equal to the change of y
over the change of x,
sometimes referred to as the rise over the run.
And now we have enough information to graph the first line.
We begin by plotting the vertical y-intercept,
which is 0,1 here.
And then because the slope is one over one,
we go up one, right one, up one, right one,
which we can continue doing over and over again.
Now if you want to find points on the left,
we could write positive one over positive one
as negative one over negative one.
Notice how this does simplify to positive one.
By using the slope in this form,
if we go back to the vertical intercept,
we would now go down one and left one,
down one and left one,
and notice how these points are still on the same line.
Let's go and sketch the line.
And now let's work on graphing the second line
given by the equation, three x minus three y equals four.
And let's go ahead and solve this for y.
So we begin by subtracting three x on both sides.
Simplifying.
We have negative three y is equal to,
let's write the right side as negative three x plus four.
And now to solve for y,
we divide both sides by negative three.
I'm going to divide every term by negative three.
And now simplifying on the left,
negative three divided by itself is equal to one.
One times y is y.
We have y equals, again, negative three divided by itself
simplifies to one giving us one x or just x.
I'm going to go and leave it as one x.
And then we have plus four divided by negative three
which gives plus negative 4/3 or minus 4/3.
Notice half of the second line,
the slope is also positive one.
If we want positive one over positive one,
and b, the vertical intercept is now negative 4/3,
which gives us the ordered pair zero comma negative 4/3.
The important thing to recognize here
is that the slope of the second line
is the same as the slope of the first line,
and the vertical or y-intercepts are different,
and therefore, the lines are going to be parallel.
They will not intersect.
So let's go and plot the vertical intercept
of zero comma negative 4/3.
Negative 4/3 is equal to negative one and 1/3.
So from the origin, go down one and 1/3 units.
Approximately here.
And the slope is one over one,
so they go up one, right one.
Up one, right one. Up one, right one, and so on.
Again, we know the lines are parallel.
Let's go ahead and sketch the second line.
Because parallel lines do not intersect,
this system has no solution.
We can also classify the system as inconsistent
because it has no solution.
I hope you found this helpful.