SECTION 1.3 Learning Objectives
1.3: Simplifying and Solving Linear Equations
- Apply the Distributive Property to simplify and then solve algebraic equations
- Combine like terms to simplify and then solve algebraic equations
- Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations
The Distributive Property
As we solve linear equations, we often need to do some work to write the linear equations in a form we are familiar with solving. This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.
Parentheses can make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.
The Distributive Property of Multiplication
For all real numbers a, b, and c, [latex]a(b+c)=ab+ac[/latex].
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.
The video below will review some examples of this distributive property in action.
Apply the Distributive Property to simplify and then solve algebraic equations
Example 1
Solve for [latex]a[/latex].
[latex]4\left(2a+3\right)=28[/latex]
Combine like terms to simplify and then solve algebraic equations
Sometimes we need to combine like terms on one or both sides of the equations to simplify before solving the equation. Like terms are terms with the same variable part. Constant terms are also considered like terms.
Example 2
Solve: [latex]7x-9-2x=-5+11[/latex]
The video below will show some more examples of combining like terms to solve an equation
Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations
In the next example, we will use both the Distributive Property and combining like terms to simply before solving the equation.
Example 3
Solve: [latex]5-14=3-2(x+1)+7x[/latex]
The video below will show some more examples of this type: