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.