 So let's see if we can find algebraic solutions to systems of equations, and the first approach we'll try is known as substitution. To solve a system of equations requires finding a value for the variables that makes all the equations true. But if all the equations are true, then we can always replace what's on one side of an equals with what's on the other side anywhere we see it. Remember, equals means replaceable. And this idea allows us to solve by substitution. So let's try and solve the system y equals 3x minus 5, y equals minus 2x plus 3. So let's start off with a true statement. y is equal to y. While this might not seem to be a particularly profound thing to write, remember that success in life and in mathematics is largely based on your ability to do bookkeeping, and part of that bookkeeping is keeping in mind where everything came from. Now equals means replaceable. So this first statement, y equals 3x minus 5, well I can replace a y with 3x minus 5. Now the crucial idea here is that I can replace this y with something, and if it's not 3x minus 5, I might actually have an equation I can solve. If only I knew something that y was equal to. Oh wait, I know y is equal to minus 2x plus 3. Equals means replaceable, so instead of writing y, I can write minus 2x plus 3. And now I have an equation that I can solve. So solving this equation gives us x equals 8 fifths. Since there are two variables x and y, finding x equals 8 fifths is only half a solution. We need to know what y is. If only I had some way of computing y once I do the value of x. Unfortunately we don't have one way to calculate y from the value of x. We actually have two ways we can calculate y. Take your pick. I'll use the first equation because that's the first equation. Equals means replaceable, so I know x equals 8 fifths, so I'll replace and evaluate. And that gives me y equals negative 1 fifth, so the solution is x equals 8 fifths, y equals negative 1 fifth. Now you might remember that when we tried to solve this system of equations by graphing, we went into the problem of not being able to read the coordinates accurately enough. And that's why we needed an algebraic method. On the other hand, remember that any time you get a supposed solution, it's always nice to check to make sure that everything is done correctly. And one of the ways we can use this graph to check is to note that our intersection point looks like it could have x coordinate 8 fifths and y coordinate negative 1 fifth. So this step is sometimes referred to as setting two equations equal to each other. But how you speak influences how you think. And when you say things like that, you set yourself up for disaster. For example, in this system, so we could set our equations equal to each other, 2x minus y equals 3x plus 2y, but that would be incorrect. Why is that? Well remember, equals means replaceable. This 2x minus y is equal to 3, so I can replace it with 3. Likewise, 3x plus 2y is the same as 5, so I can replace it with 5. And unless you're a politician who wants to insist that theirs is the biggest ever, 3 is not equal to 5. And this is why that bookkeeping step is so important. We know we can say y is equal to y, and so if I knew what y was, I can replace it. So we'll solve for y. From the first equation, we can solve for y and get y equals 2x minus 3. From the second equation, we can solve for y and get y equals 5 halves minus 3 halves x. And equals means replaceable. In every place I see a y, I can replace it with a 2x minus 3, or I can replace it with a 5 halves minus 3 halves x. So we'll replace, and that gives us an equation we can solve. So let's solve our equation. If we multiply everything by 2, that will eliminate all of our fractions. So let's do that, and solve, giving us x equals 11 sevenths as half a solution. To get the other half of the solution, we need to solve for y. We can use either equation to solve for y, since the equation on the left is the sinister one, we'll use it. We know that x is equal to 11 sevenths, equals means replaceable, so we'll replace. And while you should be able to work with fractions, remember you don't have to. If we multiply everything by the common denominator, we can get rid of the fractions. So let's multiply everything by 7. Solving for y gives us 1 seventh, so x equals 11 sevenths, y equals 1 sevenths is a solution. If we want to solve this system of equations, notice that since we already have one equation solved for y, we can substitute. So we'll take the equation that isn't solved for y, and equals means replaceable, we can replace y with 3x minus 5. It's important to remember to use parentheses, y is all of 3x minus 5, so all of 3x minus 5 should appear inside a set of parentheses. Simplifying, and most of us recognize that 5 is not equal to 8. So this statement is never true, and so this equation has no solution.