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Section 6.1 – Section 6.3 Systems of Linear Equations – Graphs and Solving A Unique Solution No Solution Infinitely Many Solutions Section 6.1 – Section 6.3 – Systems of Linear Equations – Graphs and Solving 1 Solving Linear Systems of Equations Systems of linear equations may be solved graphically or algebraically. When solving algebraically, we can use either the substitution method or elimination method. Substitution Method 1. Solve one equation for one variable. 2. Substitute this expression into the other equation for that variable solve for in step 1. This will give an equation in one variable. 3. Solve the equation. 4. Substitute back to find the value of the other variable. Example 1: Solve the given system of linear equations, then describe the solution set for the system. x + 2y = 8 2x − 4 y = 0 Elimination Method 1. Choose a variable to eliminate. Multiply one or both equations by a nonzero constant, then add the two equations. 2. Solve the resulting equation. 3. Substitute back to find the value of the other variable. Example 2: Solve the given system of linear equations, then describe the solution set for the system. 3x − 6 y = 12 − x + 2 y =−4 Section 6.1 – Section 6.3 – Systems of Linear Equations – Graphs and Solving 2 Example 3: Solve the given system of linear equations, then describe the solution set for the system. 3x − 5 y = 0 2x − 3y = 1 Example 4: Solve the given system of linear equations, then describe the solution set for the system. x + 4y = 14 2x − y = 4 Section 6.1 – Section 6.3 – Systems of Linear Equations – Graphs and Solving 3 Example 5: Solve the given system of linear equations, then describe the solution set for the system. Graph the system. = y 2x − 3 y= −3 x + 2 Section 6.1 – Section 6.3 – Systems of Linear Equations – Graphs and Solving 4