Substitution Method

Definition

The substitution method is a technique used to solve systems of linear equations by isolating one variable in one equation and then substituting that expression into the other equation(s) to find the values of the remaining variables. This method is applicable in the context of graphing linear functions, as well as solving systems of linear equations in two or three variables, and even systems of nonlinear equations and inequalities in two variables.

5 Must Know Facts For Your Next Test

1. The substitution method is particularly useful when one of the equations in the system can be easily solved for one of the variables.
2. Applying the substitution method involves isolating a variable in one equation, substituting that expression into the other equation(s), and then solving for the remaining variable(s).
3. The substitution method can be used to solve systems of linear equations with any number of variables, though it becomes more complex as the number of variables increases.
4. In the context of graphing linear functions, the substitution method can be used to determine the slope and y-intercept of a line, as well as the point of intersection between two or more lines.
5. When solving systems of nonlinear equations and inequalities in two variables, the substitution method can be used to isolate one variable and then substitute that expression into the other equation or inequality.

Review Questions

• Explain how the substitution method can be used to solve a system of linear equations in two variables.
  • To solve a system of linear equations in two variables using the substitution method, you would first isolate one variable in one of the equations, such as $x$ in the equation $x = 3y + 2$. You would then substitute this expression for $x$ into the other equation, reducing the system to a single equation in one variable (in this case, $y$). Once you have solved for $y$, you can substitute that value back into the original equation to find the value of $x$, thereby determining the solution to the system.
• Describe how the substitution method can be used to graph a system of linear equations in two variables.
  • When graphing a system of linear equations in two variables, the substitution method can be used to determine the slope and y-intercept of each line. By isolating one variable in one of the equations, you can substitute that expression into the other equation to find the slope and y-intercept. For example, if the system is $2x + 3y = 12$ and $4x - y = 8$, you could isolate $y$ in the second equation to get $y = 4x - 8$. Substituting this into the first equation gives you $2x + 3(4x - 8) = 12$, which can be simplified to $14x - 24 = 12$, allowing you to determine the slope and y-intercept of the first line. This process can be repeated for the second equation to fully characterize the system graphically.
• Analyze how the substitution method can be used to solve a system of nonlinear equations in two variables.
  • When dealing with a system of nonlinear equations in two variables, the substitution method can still be applied, though the process becomes more complex. The key is to isolate one variable in one of the equations, just as with linear systems, and then substitute that expression into the other equation(s). However, since the equations are nonlinear, the resulting equation(s) will also be nonlinear, potentially requiring the use of advanced techniques such as factoring, completing the square, or using the quadratic formula to solve for the remaining variable(s). Once the values of the variables have been determined, they can be substituted back into the original equations to verify the solution. The substitution method allows you to transform a system of nonlinear equations into a single, potentially more manageable equation, making it a valuable tool for solving these types of systems.

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 fewer variables.

Augmented Matrix: An augmented matrix is a matrix that combines the coefficients of the variables and the constants from a system of linear equations, allowing for efficient manipulation and solution using matrix operations.

Graphing Method: The graphing method for solving systems of linear equations involves plotting the equations on a coordinate plane and finding the point of intersection, which represents the solution to the system.