key term - Isolate the Variable

Definition

Isolating the variable is a fundamental algebraic technique used to solve linear equations. It involves performing mathematical operations to move all the terms containing the unknown variable to one side of the equation, while moving all the constant terms to the other side. This allows the variable to be expressed alone, making it possible to determine its value.

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.
2. To isolate the variable, you must perform inverse operations to move all the terms containing the variable to one side of the equation.
3. Common inverse operations used to isolate the variable include addition/subtraction and multiplication/division.
4. Simplifying the equation after isolating the variable can make it easier to determine the final solution.
5. Isolating the variable is a fundamental skill that is used throughout the study of algebra and beyond.

Review Questions

• Explain the general strategy for isolating the variable in a linear equation.
  • The general strategy for isolating the variable in a linear equation involves performing inverse operations to move all the terms containing the variable to one side of the equation. This typically involves using addition or subtraction to isolate the variable term, and then using multiplication or division to solve for the variable's value. For example, in the equation $2x + 5 = 17$, you would first subtract 5 from both sides to isolate the $2x$ term, resulting in $2x = 12$. Then, you would divide both sides by 2 to solve for $x$, giving you $x = 6$.
• Describe how the concept of inverse operations is used to isolate the variable in a linear equation.
  • The concept of inverse operations is crucial when isolating the variable in a linear equation. Inverse operations are mathematical operations that undo each other, such as addition and subtraction or multiplication and division. To isolate the variable, you need to perform the inverse operation to the terms on both sides of the equation to move all the variable terms to one side. For example, if the equation is $3x + 4 = 19$, you would subtract 4 from both sides to isolate the $3x$ term, resulting in $3x = 15$. Then, you would divide both sides by 3 to solve for $x$, which would give you $x = 5$. By using inverse operations, you can systematically isolate the variable and solve the equation.
• Analyze the role of simplifying an equation after isolating the variable and explain how it can aid in determining the final solution.
  • Simplifying an equation after isolating the variable can be a helpful step in determining the final solution. Once you have moved all the variable terms to one side of the equation, simplifying the expression can make it easier to solve for the variable's value. This may involve combining like terms, performing additional inverse operations, or simplifying any remaining expressions. For example, if the isolated equation is $5x = 20 + 3x$, simplifying it by subtracting $3x$ from both sides would give you $2x = 20$, which can then be easily solved by dividing both sides by 2 to find that $x = 10$. Simplifying the equation in this way can make the final step of determining the variable's value more straightforward and less prone to errors.

Related terms

Linear Equation:

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

Inverse Operations:

Inverse operations are mathematical operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division.

Simplify:

Simplifying an expression involves combining like terms and performing operations to make the expression as simple and easy to understand as possible.