key term - Isolate the Variable

Definition

Isolating the variable is the process of manipulating an equation to solve for a specific unknown variable by performing inverse operations to move all other variables and constants to one side of the equation. This technique is essential in solving linear equations and is a fundamental skill in elementary algebra.

5 Must Know Facts For Your Next Test

1. Isolating the variable is a crucial step in solving linear equations, as it allows you to determine the value of the unknown variable.
2. The addition and subtraction properties of equality are used to isolate the variable by moving all terms with the variable to one side of the equation.
3. The division and multiplication properties of equality are used to isolate the variable by dividing or multiplying both sides of the equation by the coefficient of the variable.
4. Isolating the variable often involves performing multiple inverse operations, such as first subtracting a constant from both sides, then dividing both sides by the coefficient of the variable.
5. Properly isolating the variable is essential for finding the correct solution to a linear equation and is a foundational skill in elementary algebra.

Review Questions

• Explain how the addition and subtraction properties of equality can be used to isolate the variable in a linear equation.
  • The addition and subtraction properties of equality state that you can add or subtract the same value from both sides of an equation without changing the equality. When solving a linear equation, you can use these properties to move all terms containing the variable to one side of the equation, effectively isolating the variable. For example, if you have the equation $2x + 5 = 11$, you can subtract 5 from both sides to get $2x = 6$, and then divide both sides by 2 to solve for $x$.
• Describe the role of the division and multiplication properties of equality in isolating the variable in a linear equation.
  • The division and multiplication properties of equality state that you can divide or multiply both sides of an equation by the same non-zero value without changing the equality. When solving a linear equation, you can use these properties to isolate the variable by dividing or multiplying both sides by the coefficient of the variable. For instance, if you have the equation $3x = 12$, you can divide both sides by 3 to get $x = 4$. This allows you to solve for the variable by performing the inverse operation to the coefficient.
• Analyze the step-by-step process of isolating the variable in a multi-step linear equation, such as $4x - 2 = 10 + 3x$.
  • To isolate the variable in the equation $4x - 2 = 10 + 3x$, you would first need to perform the inverse operation of subtraction to move the constant term $-2$ to the other side of the equation, resulting in $4x = 12 + 3x$. Next, you would use the inverse operation of subtraction again to isolate the variable terms on one side, giving you $x = 12$. Finally, you would divide both sides by the coefficient of the variable, which is 1 in this case, to solve for the variable $x$. This step-by-step process of applying inverse operations to isolate the variable is a fundamental skill in solving linear equations in elementary algebra.

Related terms

Inverse Operations:

Operations that undo each other, such as addition and subtraction or multiplication and division, used to isolate a variable in an equation.

Equation:

A mathematical statement that two expressions are equal, often used to represent relationships between variables and constants.

Linear Equation:

An equation in which the variable(s) appear in the first power and the graph is a straight line.