Unit 2: Linear Equations
We use equations everyday without realizing it. Examples include calculating the unit price to compare the price of brands in the grocery store, converting inches into feet (or centimeters into meters), estimating how much time it will take to drive to a destination at a certain speed.
In this unit, we explore formal procedures for solving equations. After reviewing basic math rules, we apply the skills we learned in Unit 1 to simplify the sides of an equation before attempting to solve it and work with equations that contain more than one variable. Because variables represent numbers, we use the same rules to find the specific variables we are looking for.
Completing this unit should take you approximately 5 hours.
2.1: Definition of an Equation and a Solution of an Equation
We define an equation as a statement that contains a variable, which may or may not be true, depending on the value of the variable. Solving an equation means finding the possible values of the variable that make the equation true.
2.2: Addition/Subtraction Property of Equations
When solving algebraic equations, we need to be aware of the properties of the types of mathematical operations we are doing. The first property we explore is the addition and subtraction property of equations.
2.3: Multiplication/Division Property of Equations
Much like in the previous section we must use the properties of multiplication and division when solving algebraic expressions involving these types of calculations.
2.4: Equations of the Form x + a = b and x − a = b
Algebraic equations can be categorized based on the form and types of operations in the equation. In the next few sections, we will explore different forms of equations.
The first form is the simplest: x + a = b or x − a = b. An example of this type of equation is: 5 + x = 8.
2.5: Equations of the Form ax = b and x/a = b
The next general form of equations involve multiplying or dividing the variable by a coefficient. These equations are of the form ax = b or x/a = b. An example of this type of equation is: x/2 = 6.
2.6: Equations of the Form ax + b = c
Often types of mathematical operations are combined in an equation. For example, multiplication can be combined with addition in an equation. An example of this type of equation is: 2x + 1 = 11. This requires a two-step process for solving the equation.
2.7: Equations of the Form ax + b = cx + d
This section involves solving more complicated equations where the variable appears on both sides. We can use what we learned about combining like terms to make solving these types of equations possible.
2.8: Equations with Parentheses
The last general type of linear equation we can solve are those involving parentheses. For example, we can have an equation 2(4 + x) = 12. We need to use order of operations and the distributive property to solve this type of equation.
2.9: Solving Literal Equation for One of the Variables
We can use the methods we learned in the previous sections to solve literal equations, or formulas which often have more than one variable. When a literal equation has more than one variable, we can solve for the variable of interest with respect to the other variable.
For example, consider the equation 2a + b = 10. Here, there are two variables, a and b. If we want to solve for b, we can do so with respect to a. We can subtract 2a from both sides to obtain: 10 − 2a = b.