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Notes on the Difference of 2 Squares
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Expansion of a Product of Binomials
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Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
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The Distributive Property
What is a Quadratic Equation
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Solving One-Step Equations Using Models

Objective Learn how to solve linear equations using models.

While modeling the solution steps for an equation, be sure to understand the algebra involved in each step.

Representing Equations with Models

Equations involving a single variable and integers can often be represented using models. The variable is represented by a single object and counters are used to represent the integers. The counters are shown here as either gray circles for a positive integer or black circles for a negative integer. To represent equations here using the models, we draw the same number of cups as the number of times the variable occurs, and place the appropriate number of counters on each side of the equal sign to model the integers occurring in the equation.

Example 1

x + 4 = 5

Example 2

2x + 3 = 6

Example 3

3x - 2 = 1

Solving One-Step Equations Using Models

Once the equations have been modeled, they can be manipulated by adding and removing gray or black counters to either side of the model. When there are both gray and black counters on one side, the counters can be paired up and removed. The removal of these pairs corresponds the fact that 1 + (-1) = 0, since a gray counter represents +1 and a black counter represents -1.

Example 4

Solve x + 2 = 5 for x .

Solution

First we model the equation.

Then we remove the same number of counters from each side until the cup is alone. It is critical that the same number of counters are removed from each side. To get the cup alone, we need to remove 2 gray counters from each side.

This manipulation corresponds to subtracting the number 2 from each side of the equation.

After removing counters (2 from each side), the model now looks like this.

The model above tells us that x = 3.

We can verify that this value of x is the correct solution by replacing x with it in the original equation: 3 + 2 = 5. This is correct.

Example 5

Use a model to solve x - 4 = 2.

Solution

This equation involves subtraction on one side. Subtraction of an integer is the same as the addition of the opposite integer. Therefore, the equation x - 4 = 2 can be rewritten as the equation x + ( - 4) = 2. The model for this equation is given below.

To get the cup alone in the model, we need to remove the 4 black counters that are on the left side of the equation. But since there are no black counters on the other side, we cannot use the technique shown in Example 4. In order to be able to remove the black counters, we need to add 4 gray counters to each side. As long as we add the same number and color of counters to each side, the new model will be equivalent to the original model. The model with 4 gray counters added to each side is shown below.

Now we can remove the four pairs of gray and black counters from the left side of the model. We are left with this model.

This model represents the equation x = 6. Replacing x in the original equation with 6 gives the true number sentence 6 - 4 = 2, so the solution is correct.

The following equation requires the use of more than one cup in the model. The solution of the equation is the value of x , that is, the value of just one x.

Example 6

Solve 2x = 6.

Solution

Model the equation.

Thinking of the equation as x + x = 6 is helpful at this step. There are two x s modeled on the left side of the figure, and that we can separate the counters on the right side into two groups of 3 counters each. We can then assign one group of 3 counters to each cup.

Since we are interested only in how many counters go with one x, we can focus on just one of the two parts of the previous model. The model at the bottom of the previous page shows that x = 3.

This result can be verified by substituting 3 for x in the original equation.

The process in this example corresponds to dividing each side of the equation by 2. This process is not as clear from the models as it is when we add or remove counters from each side.

 

 
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