So the focus of this will be about equations, and solving for variables.

First, we have to define what exactly is an equation. An equation is anything with two identical values on either side of an equals sign. For example:

1 = 1 Is an equation

567.26 = 567.26 Is an equation

1 + 1 = 2 Is an equation

x + x = 2x Is an equation

2*7 + 9 -7668/1337 = 17.26 Is an equation

42 Not an equation

2x^{2}+5x+9 Not an equation

2 ¹ 7 Not an equation

Now, we also need to discuss valid operations on equations. A valid operation on an equation is something that changes is so that both sides are still equal. This is usually done by doing something to both sides of the equation. For example:

5 = 5 Equal

5 – 2 = 5 – 2 Still equal

3 = 3 Still equal

Another example:

y = x + 2 Equal

y + 1 = x + 3 Still equal

y – 1 = x + 1 Still equal

2*y = 2*(x+2) Still equal

Now that that’s out of the way, let’s get into solving equations. Normally, a typical solving equation problem looks like this:

2x = 10

Solve for x

This can be really confusing at first. After all, x can mean anything!

A more accurate way to think of this problem is like this:

2x = 10

Find a value of x that makes this equation valid

With that in mind, lets loom at some of the values of x for the equation 2x = 10.

If x is… | Then the equation is… | Is it solved? |

1 | 2 = 10 | No |

2 | 4 = 10 | No |

3 | 6 = 10 | No |

4 | 8 = 10 | No |

5 | 10 = 10 | Yes |

Therefore, for the equation 2x = 10, x is equal to 5.

In general, the first step in solving any equation is getting the variable you want to solve for on one side. For example:

2x + 9 = 5x + 10 Equal

2x – 5x + 9 = 5x – 5x + 10 Still equal

-3x + 9 = 10 Still equal

-3x + 9 – 9 = 10 – 9 Still equal

-3x = 1 Still equal

Once you have all the variables on one side, apply valid operations until the equation is solved.

-3x = 1 Equal

-3x/-3 = 1/-3 Still equal

X = -1/3 Still equal

Here’s an example for two variables. Note that here we’re trying to just get x on one side, with variables still on the other side. I will discuss this in a further article.

6x + 7y + 1 = 3y + 8x + 10

6x – 8x + 7y + 1 = 3y + 8x – 8x + 10

-2x + 7y + 1 = 3y + 10

-2x + 7y – 7y + 1 = 3y – 7y + 10

-2x + 1 = -4y + 10

-2x + 1 – 1 = -4y + 10 – 1

-2x = -4y + 9 Variable is on one side

-2x/-2 = (-4y + 9)/-2

x = (-4y + 9)/-2

And that’s all there is to it! Good luck!