Algebra is the foundation of all science and technology. Algebra is the foundation of economics and engineering. Algebra is a key to understanding logic and everything in the world around us. The goal is to find the number in the red box. The red box is what we call a variable. The red box can appear anywhere in the equation. The question is, what number can we put in the box to make the equation true? All boxes with the same letter on them must have the same number inside. The question is what number we can simultaneously put into both red boxes to make the equation true? In this case, there is only one correct answer. The same number must go into the two blue boxes. A different number can go into the purple box. In this case, there are many correct answers. Life rarely has only one variable. Here, the two blue boxes are one variable. The purple box is a second variable. There are an infinite number of solutions to this problem. Variables can represent positions in space. Here we have three variables. Each can have a different number. One solution is for Y to be 2, and for Z to be 4. This represents moving 2 in the Y direction and moving 4 in the Z direction. Another solution is for Y to be 5 and for Z to be 1. The graph now shows the new solution, and the previous solution. Another solution is for X to be 3, for Y to be 2, and for Z to be 1. The graph now shows all three solutions we have found so far. Suppose we keep adding all the possible solutions to the graph. Eventually, this will be the result. If we include fractions as possible solutions, then the graph will look like this. Every equation has a graph that represents all its possible solutions. If we make a graph of all the possible solutions to this equation, then we will get the following. Now consider this equation. This is one possible solution. And so is this. Since zero multiplied by any number is zero, any values for X and Z are possible solutions, provided that Y is 2. The graph of all the possible solutions looks like this. Now let’s look at all three equations at the same time. What values for X, Y, and Z will make all three equations true? Points that are solutions to all three equations must be on all three graphs. There are only two points that are on all three graphs. Therefore, there are only two solutions. One solution is for X to be 0, for Y to be 2, and for Z to be 4. Another solution is for X to be 1, for Y to be 2, and for Z to be 3. These values will make all three equations true at the same time. We call this solving a system of equations. Here is another way to represent this solution. X weighs the same as one ball, Y weighs the same as two balls, and Z weighs the same as three balls. When the two sides weigh the same, we say that they are equal. This is still true when we replace the numbers with variables. If we make the same change to both sides of an equation, the two sides will still be equal. By making the same change to both sides of an equation, we can find the values of variables. If we add two equations together, the two sides will still be equal. We can use this to find what values will make all the equations true at the same time. We now know that Y is equal to 2. We now know that Z is equal to 3. We previously found that Y is equal to 2. Now we can find out what X is. There is another way to find out what the variables are. Now that there is only one variable, it is easier to find the solution. We now know that B is equal to 4. We can now use the first equation to find A and we can use the third equation to find C. If two variables are equal, one can replace the other. The graph where two variables are equal looks like this. If the equation is modified, the graph changes. To avoid confusing multiplication with variable X, we can use a dot for the multiplication. Or we can write it like this. As the equation is modified, the graph changes. The second green number is where the line intercepts the Y axis. The first green number is the line’s slope. This is always true for all numbers and variables. This is always true for all numbers and variables. C is the distance between the point and the center of the graph. In this case, the distance is seven. This is the equation for all the points with this distance from the center of the graph. The graph of all the solutions to this equation is a sphere. The green number is the radius of the sphere. Multiplying one of the variables squeezes the graph in that dimension. This will be true for all graphs and equations. To understand why this happens, consider a graph with just one dimension. The solutions to the equations are now squished together. Subtracting a number from the variables shifts all graphs. With this equation, each value of Z is a circle with a different radius. This is the same graph and the same equation that we saw earlier. Let’s ignore the X dimension. Subtracting a number from X shifts the graph. This is another way to write the new equation. Now let’s shift the graph in the Y direction. This is another way to write the new equation. These two points will have the following values. These are the solutions to this equation when Y is 0. This is what we call the quadratic equation. We now know how to solve most of the algebra problems we will encounter in life.