How do you solve a system of linear functions?

Solving a system of linear functions typically involves finding the values of the variables that satisfy all equations in the system simultaneously. One common method for solving such systems is the substitution method, which involves the following steps:

Steps to Solve a System of Linear Equations Using Substitution

The substitution method, as described in the provided reference, involves the following ordered steps:

1. Solve for one variable: Solve one of the equations for one of the variables. For example, isolate either x or y in one of the equations.
2. Substitute: Substitute the expression obtained in Step 1 into the other equation. This will result in an equation with only one variable.
3. Solve the resulting equation: Solve the single-variable equation obtained in Step 2 to find the value of that variable.
4. Back-substitute: Substitute the value found in Step 3 back into either of the original equations (or the equation from Step 1) to solve for the other variable.

Example

Consider the following system of equations:

• Equation 1: y = 2x + 1
• Equation 2: 3x + y = 11

1. Solve for one variable: Equation 1 is already solved for y.
2. Substitute: Substitute 2x + 1 for y in Equation 2: 3x + (2x + 1) = 11
3. Solve the resulting equation: Simplify and solve for x:
   • 5x + 1 = 11
   • 5x = 10
   • x = 2
4. Back-substitute: Substitute x = 2 into Equation 1:
   • y = 2(2) + 1
   • y = 5

Therefore, the solution to the system of equations is x = 2 and y = 5.