In this lesson, you will review how to solve an algebraic equation and expand your knowledge by solving two-step equations. You will examine how solving these equations strengthens your problem solving skill. Specifically, this lesson will cover:
1. Solving a Two-Step Linear Equation
This lesson builds upon what we learned earlier about using inverse operations to isolate the variable, applying this knowledge to equations requiring two steps to solve for the unknown value. Remember, when solving an equation for a variable, our main goal is to isolate a variable. In other words, we want to get the variable by itself on one side of the equation, with all other expressions on the other side of the equals sign. In this process, we must always remember that if we perform an operation on one side of the equation, we must do the same on the other side of the equal sign. Let's look at an example:
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To solve a two-step equation, first move the constants (numbers that don’t have variables) to one side of the equation, then isolate the variable by canceling out the coefficient (numbers multiplied or divided by the variable).
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EXAMPLE
Solve for
x in the equation
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The equation
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Again, let's get rid of the value not touching the variable. So, add 8 to both sides because it is the opposite of subtraction.
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The constant is now removed from the right side.
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Divide both sides by 2 because it is the opposite of multiplication.
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On the right side, the variable is isolated, and the solution is x = 14.
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Whatever we do on one side of the equation has to be done on the other side of the equation. This is known as the Rule of Equality.
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2. Real-World Examples
When working with word problems, it is always a good idea to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consecutive numbers or geometry problems.
IN CONTEXT
Speedy Taxi company charges $5 for the initial pick up and then $2/mile. If you spent $33, how many miles did you travel?
First, establish the variable: x = miles traveled
Now set up your equation.
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The equation.
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To solve for x, first subtract 5 from both sides because it is the opposite of addition (or, in this case, a positive number).
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The constant is now removed from the left side.
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Divide both sides by 2 because it is the opposite of multiplication.
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On the left side, the variable is isolated, and the solution is x = 14.
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You traveled 14 miles.
IN CONTEXT
A sofa and a loveseat together costs $444. The sofa costs double the loveseat. How much do they each cost?
First, establish the variable. With no information about the loveseat, this is our x: Loveseat = x.
The sofa is double the loveseat, so we multiply by x by 2 to represent the cost of the loveseat. Together, they cost $444, so we add x and 2x and create the following equation to solve:
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The equation
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Parentheses are not needed; combine like terms x and 2x.
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Divide both sides by 3.
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On the left side, the variable is isolated, and the solution is x = 148.
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The love seat costs $148, which means the sofa costs twice as much, or 2($148) = $296.
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Be careful of problems such as these. Many students see the phrase "double" and believe that it means we only have to divide 444 by 2 and get sofa equals 222 for one or both of the prices. As you can see, this will not work. By clearly labeling the variables in the original list, we know exactly how to set up and solve these problems.
Consider how solving two-step linear equations strengthen your problem solving ability—for instance, when two-step equations are somewhat complex, improving your focus during the problem solving process can you help you get to the solution.
The process of solving a two-step linear equation involves isolating the variable you want to solve for. When isolating a variable, it is helpful to have a review of inverse operations: addition and subtraction are inverse, and multiplication and division are inverse. Keep in mind when applying inverse operation that this will cancel the operations around the variable. Also, in using the inverse operations, use the order of operations in reverse order. Finally, simplifying before isolating a variable, such as distributing or combining like terms, can be helpful. These types of equations strengthen your problem solving skill by enabling you to focus on complex ideas, follow multi-step processes, and think critically to find solutions, even solutions in the real-world!.
Best of luck in your learning!
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