2.04 Equations with variables on both sides

When we solve equations with variables on both sides, we will mostly follow all the usual steps that we would when solving equations.

The only difference is that we want to rearrange the equation so that all of the variables are on the same side of the equation and all of the constants are on the other side of the equation. This process is similar to grouping like terms.

The side you choose to move variables to does not make a difference to the final solution; however, there is usually a way that will make it easier to solve. For example, if we were solving the equation: 10x - 4 = 6xif we move 6x to the left hand side of the equal sign our x-term is positive. Then we move the -4 to the right hand side and it will also be a positive number.

Examples

We are already familiar with equations that have one solution, such as x=4. However, it's also possible for an equation to have either an infinite number of solutions or no solutions.

When equations have exactly one solution, there is only one value that can be substituted for the variable that makes the equation true. The solution of an equation with exactly one solution looks like this:

x=a

When equations have no solutions, there is no value that can be substituted for the variable that makes the equation true. We will get a false statement as a result. The solution of an equation with no solutions looks like this:

a=b(where a and b are different numbers).

When equations have infinitely many solutions, any value can be substituted for the variable and make the equation true. If we try to solve the equation and get a variable or a number equal to itself, then it has infinitely many solutions. For example:

a=a

Examples