The division property of equality states that if two expressions are equal, and you divide both sides by the same non-zero number, the resulting expressions will still be equal. This property allows you to isolate variables in an equation by dividing both sides by the coefficient of the variable.
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The division property of equality allows you to divide both sides of an equation by the same non-zero number without changing the validity of the equation.
This property is particularly useful when solving for a variable that has a coefficient, as you can divide both sides by the coefficient to isolate the variable.
The division property of equality is one of the key properties of equality that, along with the multiplication property of equality, forms the foundation for solving linear equations.
Applying the division property of equality correctly is essential when solving equations that involve fractions or decimals, as it allows you to eliminate the denominator and simplify the equation.
Understanding the division property of equality is crucial for solving a wide range of equation types, including those encountered in topics such as 3.5 Solve Equations Using Integers and 8.2 Solve Equations Using the Division and Multiplication Properties of Equality.
Review Questions
Explain how the division property of equality can be used to solve an equation with a variable coefficient.
The division property of equality states that if two expressions are equal, and you divide both sides by the same non-zero number, the resulting expressions will still be equal. This property allows you to isolate a variable in an equation by dividing both sides by the coefficient of the variable. For example, if you have the equation $5x = 20$, you can divide both sides by 5 to solve for $x$, resulting in $x = 4$. The division property of equality ensures that the solution remains valid after this step.
Describe how the division property of equality is used in the context of solving equations using integers, as covered in topic 3.5.
In the topic 3.5 Solve Equations Using Integers, the division property of equality is essential for solving equations that involve integer coefficients. For instance, if you have the equation $8x = 32$, you can divide both sides by 8 to isolate the variable $x$ and find the solution. The division property of equality ensures that the equation remains valid throughout this process, allowing you to solve for the unknown variable. Understanding and correctly applying the division property of equality is a crucial skill for solving a wide range of linear equations involving integers.
Analyze how the division property of equality is used in conjunction with the multiplication property of equality to solve equations, as discussed in topic 8.2 Solve Equations Using the Division and Multiplication Properties of Equality.
The division property of equality, along with the multiplication property of equality, form the foundation for solving a variety of linear equations, as covered in topic 8.2. When solving equations, you may need to use both properties in succession to isolate the variable. For example, if you have the equation $\frac{3x}{4} = 6$, you can first multiply both sides by 4 to eliminate the fraction, resulting in $3x = 24$. Then, you can apply the division property of equality to divide both sides by 3 and solve for $x$. The combination of these two key properties of equality allows you to systematically solve complex linear equations.