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Studying for a test? Prepare with these 4 lessons on Module 4: Linear equations.

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# Systems of equations number of solutions: y=3x+1 & 2y+4=6x

Video transcript

We're told to graph this
system of equations and identify the number of solutions
that it has. And they have the system
of equations here. So they want us to graph each of
these equations and think a little bit about
the solutions. So the first equation here--
I'll rewrite it, so I'll graph it in the same color
that I write it. This first equation's already in
slope-intercept form, y is equal to 3x plus 1. We see that the slope, or m, is
equal to 3, and we see that the y-intercept here
is equal to 1. So let me be clear, that
is also the slope. I just called it m because a lot
of times people say y is equal to mx plus b. So we can graph it. We can look at its y-intercept,
the point 0, 1 must be on this graph. So that's the point
0, 1 right there. This is the y-axis, that
is the x-axis. And the slope is 3. That means if we move 1 in the
positive x-direction, we're going to move up 3 in the
positive y-direction. So we move 1 in the x-direction,
we move up 3. If we moved 2 in the
x-direction, we would move up 6. Just like that. Because 6 over 2 is still 3. Likewise, if we moved down 1, if
we went negative 1 in x, we would go negative 3 in y. So negative 1, negative 3. Because negative 3 divided
by negative 1 is still 3. If we went negative 2 in x, we
would go negative 6-- 1, 2, 3, 4, 5, 6 in y. So these are all points along
the line, and I can connect the dots now. So let me do that. So let me connect the dots
as best as I can. This should be a line,
not a curve. My hand isn't 100% steady, but I
think you get-- let me do it a little bit better than that. I think I can do a better
job than that. Let me draw-- that's
even worse. All right. Last attempt. That's throwing me off. So last attempt right here. There you go. So that's that first line
right there. y is equal to 3x plus 1. So let me do the
second one now. So it's written in standard form
right now, 2y plus 4 is equal to 6x. We want to get this in
slope-intercept form, y is equal to mx plus b. So a good place to start could
be to subtract this 4 from both sides. So it goes on the other side. So let's subtract 4 from both
sides of this equation. The left-hand side, we're left
with just a 2y, and then the right-hand side becomes
6x minus 4. So 2y is equal to 6x minus 4. And then to get everything in
terms to solve for y, we just have to divide everything
by 2. So let's divide everything
by 2, and we get y is equal to 3x minus 2. So that's the second equation
in slope-intercept form. So same drill here. The y-intercept is negative 2. So we go-- that's negative 1,
negative 2 right there, and its slope is 3. And notice its slope is the
same as the other line. So it's going to have the
same inclination. If we move 1 in the x-direction,
we move up 3 in the y-direction. 1, up 3. Just like that. If we go back 1 in
x, we go down 3. Back 1 in x, we go down 3. Just like that. So if we connect the dots
here, it'll look something like this. I'll do my best to draw
a straight line. So the second graph, 2y plus
4 equals 6, we put it into slope-intercept form
and we graphed it. Now, the whole point of this
question was to identify the number of solutions that
it has, the system. A solution to a system of
equations is an x and y value that satisfy both of
these equations. Now, if there were such an x
and y value that satisfied both of these equations, then
that x and y value would have to lie on both of
these graphs. Because this blue line is all of
the pairs of x and y's that satisfy the first equation. The red line is all of the
pairs of x's and y's that satisfy the second equation. So if something's going to
satisfy both, it's got to be on both lines. When you look here, are
there any points that are on both lines? Well, no. These two lines never
intersect. A point of intersection is a
point that is common to both of these lines. No, they don't intersect. No intersection. So there is not a solution to
this system of equations. There is no solution. We know that because these two
lines don't intersect. And you didn't even
have to graph it. The kind of giveaway was that
these are two different lines. They have different
y-intercepts, but their slopes are identical. So if you have two different
y-intercepts and your slopes are identical, then you have two
different lines that will never intersect. And if they represent a system,
or if they're the graphs of a system of
equations, that system has no solution.