SOLVING ONE-STEP EQUATIONS
PROPERTY OF EQUALITY - KEEPING A BALANCED EQUATION
So, what is the process used to solve an equation? We can think of an equation as if it were a balance scale. The two sides must be in balance at all times. For example, 5 = 5 is an equation. If you added two to one side of the equation and not the other, the equation would be "off balance." It would say 7 = 5. In other words, the two sides would no longer be equal to each other. If we add two to both sides of the equation though, we would have 7 = 7. This equation is still balanced.
division property of equality
If both sides of an equation are divided by the same value, the results are equal.
multiplication property of equality
If both sides of an equation are multiplied by the same value, the results are equal.
addition property of equality
If the same value is added to both sides of an equation, the results are equal.
subtraction property of equality
If the same value is subtracted from both sides of an equation, the results are equal.
property of equality
what happens to one side of an equation must also happen to the other side of the equation
TRANSLATING WORD PROBLEMS TO EQUATIONS
Let's go back to the problem about the concert tickets. What if the problem said that you spent $81 on concert tickets for yourself and two of your friends? Your friends are paying you back and you need to tell each one how much they owe you. How much did each ticket cost? Think About It! Why doesn't it matter which side of the equal sign the variable is on? We can write an equation: total cost = number of tickets · price of a ticket. Putting in the known values gives us $81 = 3 · p. We can simplify this equation to $81 = 3p or 3p = $81. The variable p represents the unknown, the price of a ticket. This problem differs from the previous problem because the variable is not all by itself on one side of the equation. We don't have p equal to something. We actually need to solve for the value of p. To solve an equation means to find the value of the unknown, the variable. In order to do this, we must get the variable all by itself on one side of the equal sign. It does not matter which side.
We just solved a one-step equation, because there was only one inverse operation needed to solve it. However, we solved the equation in a series of four steps. You should always follow these steps to solve a one-step equation. The steps are:
Locate the variable. Identify the operation being done to the variable. Do the inverse operation on both sides of the equation. Check the solution.
WORD PROBLEMS WITH ONE-STEP EQUATIONS Now let's solve some word problems by writing and solving one-step equations. We will need to combine the skill of translating with the skill of solving equations.
Use a variable to represent the unknown in the problem. (What are we asked to find?) Write an equation. Locate the variable in the equation. Identify the operation being done to the variable. Do the inverse operation on both sides of the equation. (The goal is to "undo" what was done to the variable.) Check the solution both in the equation and in the context of the word problem. In other words, does it make sense? State the answer to the problem.
USING INVERSE OPERATIONS TO SOLVE EQUATIONS
We want to solve the equation by solving for the unknown p. We need to get p by itself on one side of the equation. Since p is multiplied by 3, we need to have an operation that will "undo" multiplication. Division is the inverse operation, or opposite operation, of multiplication. To solve this equation, we must divide both sides by 3. When there is only one operation done to a variable that needs to be "undone," the equation is a "one-step" equation.
solution
a value or values of the variable that make an algebraic sentence true
solve
find the solution(s) to an equation
inverse operations
opposite operations that undo one another