key term - Substitution Method

Definition

The substitution method is a technique used to solve systems of linear equations by isolating one variable in one equation and substituting it into the other equation to find the values of the remaining variables. This method is particularly useful in solving systems of equations where the coefficients of the variables differ, allowing for the elimination of one variable through substitution.

5 Must Know Facts For Your Next Test

1. The substitution method involves isolating one variable in one equation and then substituting that expression into the other equation to solve for the remaining variable.
2. This method is useful when the coefficients of the variables in the system of equations differ, as it allows for the elimination of one variable through substitution.
3. The substitution method can be applied to both two-variable and multi-variable systems of equations.
4. Once the value of one variable is found, it can be substituted back into one of the original equations to find the value of the remaining variable(s).
5. The substitution method is a systematic approach that can be applied to a wide range of problems involving systems of linear equations.

Review Questions

• Explain the key steps involved in using the substitution method to solve a system of linear equations.
  • The key steps in using the substitution method to solve a system of linear equations are: 1) Isolate one variable in one of the equations, 2) Substitute the isolated expression into the other equation(s) to solve for the remaining variable(s), 3) Substitute the found value back into one of the original equations to determine the value of the other variable(s). By systematically isolating and substituting variables, the substitution method allows you to find the unique solution that satisfies all the equations in the system.
• Describe the advantages of the substitution method compared to the elimination method for solving systems of equations.
  • The main advantage of the substitution method over the elimination method is that it is particularly useful when the coefficients of the variables in the system of equations differ. In such cases, the substitution method allows for the elimination of one variable through substitution, which can be more efficient than the elimination method that requires adding or subtracting the equations to create a new equation with a single variable. Additionally, the substitution method can be applied to both two-variable and multi-variable systems of equations, making it a versatile technique for solving a wide range of linear equation systems.
• Analyze how the substitution method can be applied to solve application problems involving systems of equations, such as mixture or rate problems.
  • The substitution method is a powerful tool for solving application problems that can be represented as systems of linear equations. In mixture problems, for example, the substitution method can be used to isolate one variable, such as the amount of one ingredient, and substitute it into the other equation(s) representing the total mixture or concentration. This allows you to solve for the remaining variable(s), such as the amount of the other ingredient(s). Similarly, in rate problems involving multiple rates or distances, the substitution method can be used to eliminate one variable by isolating it in one equation and substituting it into the other equation(s) to find the values of the remaining variables. The systematic approach of the substitution method makes it well-suited for solving a variety of application problems involving systems of linear equations.

Related terms

Elimination Method:

The elimination method is another technique for solving systems of linear equations, where the goal is to eliminate one variable by adding or subtracting the equations to create a new equation with a single variable.

System of Equations:

A system of equations is a set of two or more linear equations that share the same variables and must be solved simultaneously to find the values of the variables that satisfy all the equations.

Linear Equation:

A linear equation is an equation in which the highest exponent of the variable is 1, and the variables are combined using only addition, subtraction, and multiplication.