We're asked to graph each set of equations
on the graph below, then indicate
whether there is one solution, no solution,
or infinite solutions to these systems of questions.
Let's first graph the two linear equations
by determining the intercepts.
Remember, to find the x-intercept or horizontal intercept,
we set Y equal to zero and solve for X.
And to find the vertical or y-intercept,
we set X equal to zero and solve for Y.
So beginning with the first equation
to find the X or horizontal intercept,
we will substitute zero for Y which gives us -8x.
And then minus four times zero which is zero
equals 16, solve for X, we divide
both sides by negative eight.
Simplifying, we have X equals negative two.
The horizontal or x-intercept is [-2,0], which is here.
And now let's determine the vertical
or y-intercept by setting X equal to zero.
Substituting zero for X we have negative eight times zero
which is zero, leaving us with -4y equals 16.
Solve for Y, we divide both sides by negative four.
Simplifying, we have Y equals 16
divided by negative four which is negative four.
The vertical or y-intercept is [0,-4] which is here.
And now I can graph the first line.
And now let's graph the second line
using the same technique of determining the intercepts.
Again, substituting zero for Y and solving for X
will give us the X or horizontal intercept.
And substituting zero for X and solving for Y
will give us the vertical or y-intercept.
Substituting zero for Y into the second equation,
we have -12x minus six times zero which is zero equals 24.
Solve for X, we divide both sides by negative 12.
Simplifying, we have X equals negative two.
The horizontal or x-intercept is [-2,0],
which is the same x-intercept as the first line.
And now let's find the vertical or y-intercept.
Substituting zero for X into the second equation
gives us negative 12 times zero which is zero,
leaving us with -6y equals 24.
Solve for Y, we divide both sides by negative six.
Simplifying, we have Y equals negative four.
Notice how the vertical or y-intercept is also [0,-4],
which is this point here, and therefore
we have the same line twice where the two lines coincide.
Which means the two lines have
an infinite number of points in common and therefore
the system of equations has infinite solutions.
And now for the second system of equations,
let's graph both lines by writing the equations
in slope-intercept form with the form y=mx+b.
Where b is the y-intercept and m is the slope.
So beginning with the first equation.
We need to isolate the 6y by adding 2x to both sides.
Simplifying, we have 6y equals, let's write
the right side as 2x minus six.
And that'll solve for why we divide
both sides by six or divide each term by six.
Simplifying, we have Y equals 2/6x which simplifies to 1/3x.
And then minus six divided by six gives us minus one.
And now let's do the same for the second equation
which is -x plus 3y equals zero.
The first step is to add X to both sides.
Simplifying, we have 3y equals X if we want 1x.
Solve for Y, we now divide both sides by three.
Simplifying, we have Y equals 1/3x
and if we want we can write plus zero
to recognize that the vertical intercept
or the y-intercept is zero.
Now let's go ahead and graph both lines.
Let's first graph Y equals 1/3x minus one.
The vertical or y-intercept is negative one which is here.
And the slope is the coefficient of X which is 1/3,
which means from the vertical intercept,
to find additional points, we go up one and right three.
And we can do this multiple times, up one, right three.
And now we can graph the first line.
Now let's graph the second line
given by the equation Y equals 1/3x plus zero.
So because b is zero, the vertical intercept is zero
which is the origin, which is here.
And the slope of the second line is also 1/3.
So from the origin we go up one, right three,
up one, right three and because the two lines
have the same slope and a different vertical intercept,
we can see the two lines are parallel.
So the two lines are parallel which means
they never intersect and therefore
they don't have any points in common
and therefore there is no solution
to the system of equations.
I hope you found this helpful.