Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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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 .


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.


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.


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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