key term - Variable Isolation

Definition

Variable isolation is the process of manipulating an equation to solve for a specific variable by isolating it on one side of the equation. This technique is essential in solving equations with fractions or decimals, as well as in solving applications involving systems of equations and mixture problems.

5 Must Know Facts For Your Next Test

1. Variable isolation is crucial for solving equations with fractions or decimals, as it allows you to get the variable by itself on one side of the equation.
2. In the context of systems of equations, variable isolation is used to solve for one variable in one equation and then substitute that value into the other equation to solve for the remaining variable.
3. For mixture applications with systems of equations, variable isolation is used to set up the equations and then solve for the unknown quantities, such as the amounts of different components in the mixture.
4. Inverse operations, such as addition and subtraction or multiplication and division, are used to isolate the variable by performing the opposite operation on both sides of the equation.
5. Simplifying the equation before attempting to isolate the variable can make the process easier and reduce the chances of making mistakes.

Review Questions

• Explain how variable isolation is used to solve equations with fractions or decimals.
  • To solve equations with fractions or decimals using variable isolation, you first need to identify the variable you want to isolate. Then, you can use inverse operations, such as multiplying or dividing both sides of the equation, to get the variable by itself on one side. This may involve simplifying the equation by combining like terms or removing any unnecessary operations. The goal is to manipulate the equation to have the variable of interest on one side and all other terms on the other side.
• Describe how variable isolation is used to solve applications with systems of equations.
  • When solving applications with systems of equations, variable isolation is a key technique. First, you need to set up the system of equations based on the given information. Then, you can use variable isolation to solve for one variable in one equation, typically by performing inverse operations. Once you have the value of one variable, you can substitute it into the other equation to solve for the remaining variable. This process of isolating variables and substituting values is repeated until all the unknown quantities in the system are determined.
• Analyze how variable isolation is applied to solve mixture applications with systems of equations.
  • In mixture applications with systems of equations, variable isolation is essential for determining the unknown quantities, such as the amounts of different components in the mixture. First, you need to set up the system of equations based on the given information about the mixture, such as the total volume or weight and the proportions of the components. Then, you can use variable isolation to solve for one variable in one equation, typically by performing inverse operations. Once you have the value of one variable, you can substitute it into the other equation to solve for the remaining variable. This process of isolating variables and substituting values is repeated until all the unknown quantities in the mixture problem are determined.

Related terms

Inverse Operations:

Inverse operations are mathematical operations that undo each other, such as addition and subtraction or multiplication and division. These are used to isolate a variable in an equation.

Simplifying Equations: Simplifying equations involves reducing the number of terms and removing any unnecessary operations or variables to make the equation easier to solve.

Substitution Method:

The substitution method is a technique for solving systems of equations by isolating a variable in one equation and substituting it into the other equation to solve for the remaining variable.