Solving Equations I Lesson 7.03

After completing this lesson, you will be able to say: I can define and apply inverse operations of addition or subtraction. I can solve one-variable equations containing addition or subtraction. I can solve real-world problems by writing and solving equations.

Writing Equations You already know how to translate expressions, and translating real-world problems into equations uses a similar skill set. Just locate the words that indicate the equal sign. "some number plus 20 is 55" is an example of the verbal form of the equation x + 20 = 55. The verb "is" tells you where to place the equal sign. Other mathematical words that indicate an equal sign are "equals," "yields," and "results in." Be sure to add them to your list of common action phrases

Solutions to Equations You just solved an algebraic equation using one strategy; however, there are other strategies you can use.

Solutions to Equations You can use fact families to rewrite equations Because the equation is x + 20 = 55, you can create three other equations that have the same exact meaning using fact families. Then, you can use the equation that lets you calculate x directly x = 5555 – x = 2055 – 20 = x For this problem, 55 − 20 = x solves for x directly because it is set up to equal x only. You can simply subtract to determine 35 is the solution. So, x = 35.

Solutions to Equations A bar model is a great way to see an equation visually. It does not actually solve the equation for you, but it does help you understand how each value is related in the equation. Here is how to set up a bar model. The top bar is the whole or total amount of the problem. The bottom bar is the two parts that add up to the whole. Below is a bar model for the equation x + 20 = 55. You can see that 20 and the unknown value together make 55.

Balancing an Equation An equation is similar to a scale. Think about the equation = 6. The expressions on each side of the equal sign are like the objects in the left and right trays of the scale. Because and 6 represent the same amount, the equation is "balanced." So, = 6 is a true equation.

Using a Balance Scale to solve an Equation Use the balance scale to determine the solution to the equation x + 3 = 5. A bag plus 3 blocks has the same weight as 5 blocks. What can you do to both sides of the scale to determine the weight of the bag only?

Using a Balance Scale to solve an Equation In the equation x + 3 = 5, the variable x represents the unknown weight of the bag. The goal is to isolate the variable so that it is alone on the left side of the equation. For the bag to be alone, you must remove 3 blocks from the left side. If you remove 3 blocks from the left side, you must then also remove 3 blocks from the right side to keep the balance. As you can see, the weight of the bag is the same as 2 blocks. So, x = 2.

Try it Use the balance scale to determine the solution to the equation x + 4 = 7. A bag plus 4 blocks has the same weight as 7 blocks. What can you do to both sides of the scale to determine the weight of the bag only?

Check your work You can isolate the bag by removing 4 blocks from each side, which leaves the bag on the left side and 3 blocks on the right side. The scale is balanced, so the bag must weigh the same as 3 blocks. The solution to the equation x + 4 = 7 is x = 3.

Try it Use the fact families to determine the solution to the equation x − 6 = 11.

Check your work The equation x − 6 = 11 can be rewritten in three other ways: x − 11 = 6 x = x = There are two equations you can use to calculate the value of x. Therefore, x = 17. Check the solution with substitution: x − 6 = − 6 = = 11

Solving with Opposites You can also use inverse operations to solve an equation. Inverse Operation: An operation that reverses the effect of another operation; for example, adding 3 and subtracting 3 are inverse operations. When you have an operation and an inverse operation together, they will “undo” each other.

Golden Rule for Solving Equations What you do to one side of an Equation, you MUST do to the other side of the equation. To solve an equation you must isolate the variable on one side of the equal sign by using inverse operations. This results in zero by undoing the operation

Example Solve the equation x − 7 = 18 to determine the value of x. To determine the value of x, you must isolate it on one side of the equation. Use inverse operations to get the variable alone. OperationsExplanation x − 7 = 18 x − = Use the inverse operation of subtraction by adding 7 to both sides of the equation and simplify. X = 25The solution to the equation is 25.

Try it!

Check your work

Equations in the Real World Using inverse operations is a very helpful technique for solving algebraic equations. You can use your new knowledge to solve real-world problems. Bar models can help you write an equation before you have to solve it.

Example Jackson receives a $100 gift card. If he purchases the $20 jacket and the $35 video game, how much money does he have left on the card? Go through the steps to see how equations can help Jackson. Use a bar model Jackson’s situation can be represented using the bar model below. The whole is 100 and there are three parts: the jacket, which is $20; the video game, which is $35; and the remaining balance, which is unknown.

Example Create the equation The verbal expression, "20 plus 35 plus some number equals 100," describes the bar model. Therefore, the equation can be represented as x = 100, where x is the amount of money left on the gift card. Solve the equation Now use inverse operations to solve the equation x = x = 100 First, simplify the equation using addition. 55 − 55 + x = 100 − 55 Next, use inverse operation for addition. x = 100 − 55 Simplify. x = 45

Try it Tyler has $40 to spend. He buys lunch for $11.34 and a shirt with the remaining money. How much does Tyler spend on the shirt?

Check your work Whole: 40 Parts: and x The variable x is the amount of money spent on the shirt. A verbal model of the problem is "11.34 plus some number equals 40." The algebraic equation for the problem is x = 40. Use inverse operations to determine the value of x.

Now that you completed this lesson, you should be able to say: I can define and apply inverse operations of addition or subtraction. I can solve one-variable equations containing addition or subtraction. I can solve real-world problems by writing and solving equations.