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We're told to graph this system of equations and

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identify the number of solutions that it has.

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And they have the system of equations here.

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So they want us to graph each of these equations and think a

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little bit about the solutions.

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So the first equation here-- I'll rewrite it, so I'll graph

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it in the same color that I write it.

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This first equation's already in slope-intercept form, y is

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equal to 3x plus 1.

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We see that the slope, or m, is equal to 3, and we see that

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the y-intercept here is equal to 1.

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So let me be clear, that is also the slope.

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I just called it m because a lot of times people say y is

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equal to mx plus b.

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So we can graph it.

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We can look at its y-intercept, the point 0, 1

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must be on this graph.

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So that's the point 0, 1 right there.

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This is the y-axis, that is the x-axis.

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And the slope is 3.

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That means if we move 1 in the positive x-direction, we're

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going to move up 3 in the positive y-direction.

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So we move 1 in the x-direction, we move up 3.

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If we moved 2 in the x-direction, we

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would move up 6.

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Just like that.

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Because 6 over 2 is still 3.

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Likewise, if we moved down 1, if we went negative 1 in x, we

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would go negative 3 in y.

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So negative 1, negative 3.

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Because negative 3 divided by negative 1 is still 3.

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If we went negative 2 in x, we would go negative 6-- 1, 2, 3,

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4, 5, 6 in y.

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So these are all points along the line, and I can connect

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the dots now.

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So let me do that.

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So let me connect the dots as best as I can.

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This should be a line, not a curve.

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My hand isn't 100% steady, but I think you get-- let me do it

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a little bit better than that.

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I think I can do a better job than that.

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Let me draw-- that's even worse.

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All right.

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Last attempt.

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That's throwing me off.

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So last attempt right here.

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There you go.

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So that's that first line right there. y is

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equal to 3x plus 1.

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So let me do the second one now.

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So it's written in standard form right now, 2y plus 4 is

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equal to 6x.

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We want to get this in slope-intercept form, y is

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equal to mx plus b.

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So a good place to start could be to subtract this 4 from

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both sides.

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So it goes on the other side.

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So let's subtract 4 from both sides of this equation.

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The left-hand side, we're left with just a 2y, and then the

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right-hand side becomes 6x minus 4.

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So 2y is equal to 6x minus 4.

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And then to get everything in terms to solve for y, we just

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have to divide everything by 2.

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So let's divide everything by 2, and we get y is

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equal to 3x minus 2.

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So that's the second equation in slope-intercept form.

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So same drill here.

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The y-intercept is negative 2.

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So we go-- that's negative 1, negative 2 right there, and

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its slope is 3.

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And notice its slope is the same as the other line.

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So it's going to have the same inclination.

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If we move 1 in the x-direction, we move up 3 in

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the y-direction.

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1, up 3.

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Just like that.

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If we go back 1 in x, we go down 3.

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Back 1 in x, we go down 3.

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Just like that.

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So if we connect the dots here, it'll look

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something like this.

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I'll do my best to draw a straight line.

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So the second graph, 2y plus 4 equals 6, we put it into

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slope-intercept form and we graphed it.

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Now, the whole point of this question was to identify the

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number of solutions that it has, the system.

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A solution to a system of equations is an x and y value

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that satisfy both of these equations.

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Now, if there were such an x and y value that satisfied

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both of these equations, then that x and y value would have

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to lie on both of these graphs.

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Because this blue line is all of the pairs of x and y's that

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satisfy the first equation.

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The red line is all of the pairs of x's and y's that

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satisfy the second equation.

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So if something's going to satisfy both, it's got to be

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on both lines.

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When you look here, are there any points

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that are on both lines?

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Well, no.

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These two lines never intersect.

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A point of intersection is a point that is common to both

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of these lines.

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No, they don't intersect.

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No intersection.

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So there is not a solution to this system of equations.

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There is no solution.

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We know that because these two lines don't intersect.

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And you didn't even have to graph it.

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The kind of giveaway was that these are two different lines.

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They have different y-intercepts, but their slopes

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are identical.

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So if you have two different y-intercepts and your slopes

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are identical, then you have two different lines that will

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never intersect.

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And if they represent a system, or if they're the

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graphs of a system of equations, that

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system has no solution.

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