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We're told to solve and graph the solution for the system of

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equations right here.

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And the first thing that jumps out at me, is that we might be

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able to eliminate one of the variables.

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And if we just focus on the x, we have a 4x here and we have

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a 2x right here.

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If we were to just add them right now, we would get a 6x.

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So that wouldn't eliminate it.

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But if we can multiply this 2x by negative 2, it'll become a

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negative 4x, and then when you add it, they would cancel out.

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So let's multiply this equation, this second

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equation, by negative 2.

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So I'm going to multiply both sides of this equation by

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negative 2.

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And the whole motivation is so that this 2x becomes a

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negative 4x.

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And, of course, I can't just multiply only the 2x.

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Anything I do to the left-hand side of the equation I have to

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do to every term, and I have to do to both

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sides of the equation.

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So the second equation becomes negative 4x-- that's negative

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2 times 2x-- plus-- we have negative 2 times negative y--

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which is plus 2y is equal to 2.5 times negative 2, is equal

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to negative 5.

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I just rewrote the second equation, multiplying both

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sides by negative 2.

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Now, this top equation-- I'll write it on the bottom now--

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we have 4x minus 2y is equal to positive 5.

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And now we can eliminate it.

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We can say, hey, look, the negative 4x and the positive

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4x should cancel out, or they will cancel out.

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So let's add these two equations.

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Let's add the left side to the left side, the right side to

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the right side, and we can do that because these

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two things are equal.

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We're doing the same thing to both sides of the equation.

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So what do we get?

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If we take our negative 4x plus our 4x, well, those

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cancel out.

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So you're left with nothing.

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Maybe I could write a 0 there.

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0x if you want.

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And then you have your plus 2y and your negative 2y.

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Those also cancel out.

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So you're also left with 0y.

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And then that equals negative 5 plus 5 is equal to 0.

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So this just simplifies to 0 equals 0, which is true, but

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it's kind of bizarre.

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We had all these x's and y's.

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Everything canceled out.

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So let's explore this a little bit more.

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Let's graph it and see what this 0 equals 0 is telling us

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when we try to solve this system of equations.

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So let me graph this top guy.

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I'll do it in blue.

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So right now it's in standard form.

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Let's put it in slope-intercept form.

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So we have 4x minus 2y is equal to 5.

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Let's subtract 4x from both sides.

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I want the x terms on the right-hand side.

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So then I'm left with negative 2y is equal to

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negative 4x plus 5.

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Now we can divide both sides by negative 2.

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And we are left with y is equal to positive 2x, right,

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that's positive 2x, minus 2.5.

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So let's graph that.

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The y-intercept is negative 2.5.

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So negative 2.5 right there, and then it has a slope of 2.

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So if we move up 1, if we move up in the x-direction, if we

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move to the right 1 in the positive x-direction, we

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will move up 2.

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So 1, 2.

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Right there.

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And if we were to do it again, we move up 1, 2.

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Just like that.

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So the line's going to look something like this.

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

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This is the hardest part about a lot of these

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problems. There you go.

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So that's the top equation.

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Now, let me draw the bottom equation.

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Let me draw and I'll do it in this green color.

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So this bottom equation was 2x minus y is equal to 2.5.

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And we can subtract 2x from both sides.

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The left-hand side becomes negative y is equal to 2x

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plus-- or is equal to negative 2x plus 2.5.

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Now let's multiply or divide both sides by negative 1.

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And you get y is equal to positive 2x minus 2.5.

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And let's try to graph this, and you already might notice

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something interesting about these two equations.

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You try to graph this, the y-intercept is at negative

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2.5, right there.

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The slope is 2.

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So it's going to be this exact same line.

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And you saw that algebraically.

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I didn't have to graph it.

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These two lines have the exact same equation when you put

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them in slope-intercept form.

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That's the first equation.

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That's the second equation.

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So what this 0 equals 0 is telling us is actually that

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these are the same line.

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That these actually have an infinite number of solutions.

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Any point on this line, which is both of those lines, will

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satisfy both of these equations.

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You give me an arbitrary y, solve for x in the top

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equation, that x and y will also

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satisfy the bottom equation.

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So this actually has an infinite number of solutions.

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These are the same line.

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