Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[/latex].

Solving linear equations with variables on both sides of the equation

To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, [latex]4x[/latex] or [latex]2x[/latex], however, to avoid negative coefficients, you can move the smaller term.

In this video, we show an example of solving equations that have variables on both sides of the equal sign.

Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations with variables on both sides of the equation

Recall the Distributive Property from last section

What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced to isolate the variable and solve the equation.

In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property (that we learned last section) twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!

In the following video, we solve another multi-step equation with two sets of parentheses.

In the next example, we will use both the Distributive Property and combining like terms to simplify before solving the equation.

Classifying solutions to Linear Equations

There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that do not have any solutions and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next example.

Equations with No Solutions

Notice in the example above you did not find a value for x. Solving for x the way you know how, you arrive at the false statement [latex]4=5[/latex]. Surely [latex]4[/latex] cannot be equal to [latex]5[/latex]!

This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by [latex]2[/latex] and add [latex]4[/latex] you would never get the same answer as when you multiply that same number by [latex]2[/latex] and add  [latex]5[/latex]. Since there is no value of x that will ever make this a true statement, the solution to the equation above is “no solution.”

Be careful that you do not confuse the solution [latex]x=0[/latex] with “no solution.” The solution [latex]x=0[/latex] means that the value [latex]0[/latex] satisfies the equation, so there is a solution. “No solution” means that there is no value, not even [latex]0[/latex], which would satisfy the equation.

Also, be careful not to make the mistake of thinking that the equation [latex]4=5[/latex] means that [latex]4[/latex] and [latex]5[/latex] are values for x that are solutions. If you substitute these values into the original equation, you’ll see that they do not satisfy the equation. This is because there is truly no solution—there are no values for x that will make the equation [latex]12+2x–8=7x+5–5x[/latex] true.

Algebraic Equations with an Infinite Number of Solutions

You have seen that if an equation has no solution, you end up with a false statement instead of a value for x. It is possible to have an equation where any value for x will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[/latex] and [latex]-4x[/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.

In that last example, we arrived at the true statement “[latex]3=3[/latex].” When you end up with a true statement like this, it means that the solution to the equation is “all real numbers.” Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\dfrac{3}{4}[/latex] and it will also check!

This equation happens to have an infinite number of solutions. Any value for x that you can think of will make this equation true. When you think about the context of the problem, this makes sense—the equation [latex]x+3=3+x[/latex] means “some number plus [latex]3[/latex] is equal to [latex]3[/latex] plus that same number.” We know that this is always true—it is the commutative property of addition!

In the following video, we show more examples of attempting to solve a linear equation with either no solution or many solutions.

In the following video, we show more examples of solving linear equations with parentheses that have either no solution or many solutions.