Systems of equations with elimination: 4x-2y=5 & 2x-y=2.5

We're told to solve and graph the solution for the system of equations right here. And the first thing that jumps out at me, is that we might be able to eliminate one of the variables. And if we just focus on the x, we have a 4x here and we have a 2x right here. If we were to just add them right now, we would get a 6x. So that wouldn't eliminate it. But if we can multiply this 2x by negative 2, it'll become a negative 4x, and then when you add it, they would cancel out. So let's multiply this equation, this second equation, by negative 2. So I'm going to multiply both sides of this equation by negative 2. And the whole motivation is so that this 2x becomes a negative 4x. And, of course, I can't just multiply only the 2x. Anything I do to the left-hand side of the equation I have to do to every term, and I have to do to both sides of the equation. So the second equation becomes negative 4x-- that's negative 2 times 2x-- plus-- we have negative 2 times negative y-- which is plus 2y is equal to 2.5 times negative 2, is equal to negative 5. I just rewrote the second equation, multiplying both sides by negative 2. Now, this top equation-- I'll write it on the bottom now-- we have 4x minus 2y is equal to positive 5. And now we can eliminate it. We can say, hey, look, the negative 4x and the positive 4x should cancel out, or they will cancel out. So let's add these two equations. Let's add the left side to the left side, the right side to the right side, and we can do that because these two things are equal. We're doing the same thing to both sides of the equation. So what do we get? If we take our negative 4x plus our 4x, well, those cancel out. So you're left with nothing. Maybe I could write a 0 there. 0x if you want. And then you have your plus 2y and your negative 2y. Those also cancel out. So you're also left with 0y. And then that equals negative 5 plus 5 is equal to 0. So this just simplifies to 0 equals 0, which is true, but it's kind of bizarre. We had all these x's and y's. Everything canceled out. So let's explore this a little bit more. Let's graph it and see what this 0 equals 0 is telling us when we try to solve this system of equations. So let me graph this top guy. I'll do it in blue. So right now it's in standard form. Let's put it in slope-intercept form. So we have 4x minus 2y is equal to 5. Let's subtract 4x from both sides. I want the x terms on the right-hand side. So then I'm left with negative 2y is equal to negative 4x plus 5. Now we can divide both sides by negative 2. And we are left with y is equal to positive 2x, right, that's positive 2x, minus 2.5. So let's graph that. The y-intercept is negative 2.5. So negative 2.5 right there, and then it has a slope of 2. So if we move up 1, if we move up in the x-direction, if we move to the right 1 in the positive x-direction, we will move up 2. So 1, 2. Right there. And if we were to do it again, we move up 1, 2. Just like that. So the line's going to look something like this. I'll try my best to draw a straight line. This is the hardest part about a lot of these problems. There you go. So that's the top equation. Now, let me draw the bottom equation. Let me draw and I'll do it in this green color. So this bottom equation was 2x minus y is equal to 2.5. And we can subtract 2x from both sides. The left-hand side becomes negative y is equal to 2x plus-- or is equal to negative 2x plus 2.5. Now let's multiply or divide both sides by negative 1. And you get y is equal to positive 2x minus 2.5. And let's try to graph this, and you already might notice something interesting about these two equations. You try to graph this, the y-intercept is at negative 2.5, right there. The slope is 2. So it's going to be this exact same line. And you saw that algebraically. I didn't have to graph it. These two lines have the exact same equation when you put them in slope-intercept form. That's the first equation. That's the second equation. So what this 0 equals 0 is telling us is actually that these are the same line. That these actually have an infinite number of solutions. Any point on this line, which is both of those lines, will satisfy both of these equations. You give me an arbitrary y, solve for x in the top equation, that x and y will also satisfy the bottom equation. So this actually has an infinite number of solutions. These are the same line.