1. Linear & Nonlinear Systems of Equations MATH 109 - Precalculus S. Rook

2. Overview Section 7.1 in the textbook: – Solving systems of equations by substitution 2

3. Solving Systems of Equations by Substitution

4. 4 Verifying a Solution to a System Graphically, when asked to solve a system of equations, we are looking for the point(s) of intersection of the functions Any coordinate (x, y) that satisfies BOTH equations is known as the solution of the system Two common methods to solve a system: – Substitution (this section) – Elimination (next section)

5. 5 Solving Systems of Equations by Substitution Useful when the coefficient of a variable is 1 or -1 To solve a system of equations by substitution: – Solve one equation for one of the variables – Substitute (replace) into the OTHER equation DO NOT substitute into the equation that was just used to solve for a variable! – What results is an equation in 1 variable – After solving this equation, we now have a partial coordinate – Evaluate the partial coordinate into either equation to get the full coordinate Often easier to plug into the substitution

6. Solving Linear Systems of Equations by Substitution (Example) Ex 1: Solve the system by substitution: a) b) 6

7. Solving Systems of Non-Linear Equations by Substitution (Example) Ex 2: Solve the system by substitution: a) b) 7

8. 8 Inconsistent & Dependent Systems Inconsistent systems: a system of equations which has NO solution – Graphically, the equations will never intersect Dependent systems: a system of equations which has an INFINITE number of solutions – Graphically, the equations lie on top of each other – “All real numbers” does NOT apply – Use a parameter variable to express the solution Will get a better idea after an example

9. Inconsistent & Dependent Systems (Example) Ex 3: Solve the system by substitution: a) b) c) 9

10. Summary After studying these slides, you should be able to: – Solve linear and non-linear systems of equations by substitution Additional Practice – See the list of suggested problems for 7.1 Next lesson – Two-Variable Linear Systems (Section 7.2) 10