First, let’s define some important terminology:

• variables:  variables are symbols that stand for an unknown quantity, they are often represented with letters, like x, y, or z.
• coefficient: Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3x is 3.
• term: a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[/latex] are all terms
• expression: groups of terms connected by addition and subtraction.  [latex]2x^2-5[/latex] is an expression
• equation:  an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning “the same as.” Some examples of equations are [latex]y = mx +b[/latex],  [latex]\frac{3}{4}r = v^{3} - r[/latex], and  [latex]2(6-d) + f(3 +k) = \frac{1}{4}d[/latex]

The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[/latex], the variable is [latex]x[/latex], a coefficient is [latex]10[/latex], a term is [latex]10x[/latex], an expression is [latex]2x-3^2[/latex].

Equation made of coefficients, variables, terms and expressions.

Solve algebraic equations using the Addition Property of Equality

An important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation “balanced.” If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.

Let’s look at a simple numeric equation, [latex]3+7=10[/latex], to explore the idea of an equation as being balanced.

The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let’s see what happens when 5 is added to each side.

[latex]3+7+5=10+5[/latex]

Since each expression is equal to 15, you can see that adding 5 to each side of the original equation resulted in a true equation. The equation is still “balanced.”

On the other hand, let’s look at what would happen if you added 5 to only one side of the equation.

[latex]\begin{array}{r}3+7=10\\3+7+5=10\\15\neq 10\end{array}[/latex]

Adding 5 to only one side of the equation resulted in an equation that is false. The equation is no longer “balanced,” and it is no longer a true equation!

When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.

When the equation involves addition or subtraction, use the inverse operation to “undo” the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0, the additive identity.

In the following simulation, you can adjust the quantity being added or subtracted to each side of an equation to see how important it is to perform the same operation on both sides of an equation when you are solving.

In the following video two examples of using the addition property of equality are shown.

Since subtraction can be written as addition (adding the opposite), the addition property of equality can be used for subtraction as well. So just as you can add the same value to each side of an equation without changing the meaning of the equation, you can subtract the same value from each side of an equation.

It is always a good idea to check your answer whether you are requested to or not.

The following video presents two examples of using the addition property of equality when there are negative integers in the equation.

Solve algebraic equations using the Multiplication Property of Equality

Just as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let’s look at a numeric equation, [latex]5\cdot3=15[/latex], to start. If you multiply both sides of this equation by 2, you will still have a true equation.

[latex]\begin{array}{r}5\cdot 3=15\,\,\,\,\,\,\, \\ 5\cdot3\cdot2=15\cdot2 \\ 30=30\,\,\,\,\,\,\,\end{array}[/latex]

This characteristic of equations is generalized in the Multiplication Property of Equality.

When the equation involves multiplication or division, you can “undo” these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.

If the equation involves division, you can multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1. See the next problem to see an example of this.