key term - Multiplication Property of Equality

Definition

The multiplication property of equality states that if two expressions are equal, then multiplying both sides of the equation by the same non-zero number will result in two new expressions that are also equal. This property allows us to isolate a variable in a linear equation by performing the same operations on both sides of the equation.

5 Must Know Facts For Your Next Test

1. The multiplication property of equality states that if $a = b$, then $kа = kb$ for any non-zero constant $k$.
2. This property allows us to isolate a variable in a linear equation by multiplying both sides of the equation by the reciprocal of the coefficient of the variable.
3. Applying the multiplication property of equality can help simplify linear equations and make them easier to solve.
4. The multiplication property of equality is a fundamental tool in solving linear equations, as it enables us to perform the same operations on both sides of the equation to isolate the variable.
5. Understanding and correctly applying the multiplication property of equality is crucial for solving a wide range of linear equations encountered in intermediate algebra.

Review Questions

• Explain how the multiplication property of equality can be used to solve a linear equation.
  • The multiplication property of equality states that if two expressions are equal, then multiplying both sides of the equation by the same non-zero number will result in two new expressions that are also equal. This property can be used to solve linear equations by isolating the variable. For example, if we have the equation $3x + 5 = 17$, we can multiply both sides by the reciprocal of the coefficient of $x$, which is $1/3$. This will result in the equation $x + 5/3 = 17/3$, where the variable $x$ is now isolated on one side of the equation, making it easier to solve for the unknown value.
• Describe how the multiplication property of equality is related to the concept of solving linear equations using a general strategy.
  • The multiplication property of equality is a crucial component of the general strategy for solving linear equations. This property allows us to perform the same operations on both sides of the equation to isolate the variable. By multiplying both sides of the equation by the same non-zero number, we can eliminate coefficients or move terms to one side of the equation, making it easier to solve for the unknown variable. The ability to apply the multiplication property of equality is essential in implementing the step-by-step process of solving linear equations, which often involves simplifying the equation, isolating the variable, and then solving for the unknown value.
• Analyze how the multiplication property of equality contributes to the overall understanding and mastery of solving linear equations in the context of intermediate algebra.
  • The multiplication property of equality is a fundamental concept that underpins the ability to solve linear equations in intermediate algebra. By understanding and applying this property, students can effectively isolate variables and simplify equations, which are critical skills for solving a wide range of linear equations. The multiplication property allows students to perform the same operations on both sides of an equation, maintaining the equality and enabling them to systematically work towards the solution. Mastering the application of this property is not only essential for solving individual linear equations but also lays the foundation for more advanced algebraic concepts and problem-solving techniques. Proficiency in using the multiplication property of equality is a key indicator of a student's overall understanding and ability to succeed in intermediate algebra.

Related terms

Linear Equation:

A linear equation is an algebraic equation in which the highest exponent of the variable is 1. Linear equations have the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Isolating a Variable: Isolating a variable in an equation involves performing operations to move all the terms containing the variable to one side of the equation and all the constant terms to the other side, resulting in an equation with the variable by itself on one side.

Solving Linear Equations: Solving a linear equation involves using the properties of equality, such as the multiplication property, to isolate the variable and find the value that satisfies the equation.