3.1 Solve Equations Using the Subtraction and Addition Properties of Equality
3.1 Solve Equations Using the Subtraction and Addition Properties of Equality
Learning Objectives
By the end of this section, you will be able to:
- Solve equations using the Subtraction and Addition Properties of Equality
- Solve equations that need to be simplified
- Translate an equation and solve
- Translate and solve applications
We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problem-solving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.
Solve Equations Using the Subtraction and Addition Properties of Equality
Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.
Solution of an Equation
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
The steps to determine if a value is a solution to an equation are listed here.
HOW TO: Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
EXAMPLE 1
Determine whether is a solution for .
Solution
| Substitute for y | |
| Multiply. | |
| Add. |
Since results in a true equation, is a solution to the equation .
TRY IT 1.1
Is a solution for
Show answer
no
TRY IT 1.2
Is a solution for
Show answer
no
In that section,we will model how the Subtraction and Addition Properties work and then we will apply them to solve equations.
Subtraction Property of Equality
For all real numbers , and , if , then .
Addition Property of Equality
For all real numbers , and , if , then .
When you add or subtract the same quantity from both sides of an equation, you still have equality.
We will introduce the Subtraction Property of Equality by modeling equations with envelopes and counters. (Figure .1) models the equation .
The goal is to isolate the variable on one side of the equation. So we ‘took away’ from both sides of the equation and found the solution .
Some people picture a balance scale, as in (Figure .2), when they solve equations.
The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.
Let’s see how to use Subtraction and Addition Properties of Eq