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.
Upon successful completion of this unit, you will be able to:
- determine whether a given real number is a solution of an equation;
- simplify equations using addition and multiplication properties;
- find the solution of a given linear equation with one variable;
- determine the number of solutions of a given linear equation in one variable;
- solve a literal equation for the given variable; and
- rearrange formulas to isolate a quantity of interest.
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.
Read the "Define Linear Equations in One Variable" and "Solutions to Linear Equations in One Variable" sections. Then, complete exercises 1 to 5 and check your answers.
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.
Read up to the "Solve Equations that Require Simplification" section. Pay attention to the "Solve Equations Using the Subtraction and Addition Properties of Equality" section, which gives a good example of how the two sides of an equation must be equal. After you read, complete examples 2.2 through 2.5 and check your answers.
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.
Read up to the "Sole Equations that Require Simplification" section. Complete examples 2.13 to 2.17.
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.
Watch this video for examples of these types of equations.
After you watch, complete this assessment to test yourself.
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.
- Watch these videos for a few examples of how to solve algebraic equations involving multiplication and division. Pay attention to the problem-solving steps for fractional coefficients in the third video. Instead of dividing both sides by the fraction, you can multiply both sides of the equation by the reciprocal of the fraction. It is often easier to multiply fractions rather than dividing them, so this trick can be useful.
After you watch, complete this assessment and check your answers.
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.
Watch this video for examples of how to solve these types of problems in a two-step process.
After you watch, complete this assessment and check your answers.
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.
Watch these videos to see examples of how we use like terms to solve these types of equations.
After you watch, complete this assessment and check your answers.
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.
Watch these videos to see examples of how this type of equation can be solved.
After you watch, complete this assessment and check your answers.
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.
Read the section on linear literal equations. Be sure to go through the examples in detail. After read, complete the exercises for literal equations and check your answers.
We can apply these concepts to known formulas, such as formulas for area of a shape or rates.
Watch these videos for real examples of using formulas. In the first video, the formula for perimeter of a rectangle is solved for the width. In the second, a formula is used to convert between Fahrenheit and Celsius.