Copyright © Cengage Learning. All rights reserved. 9 Systems of Equations and Inequalities Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 9.1 LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS Copyright © Cengage Learning. All rights reserved.

What You Should Learn Use the method of substitution to solve systems of linear equations in two and three variables. Use the method of substitution to solve systems of nonlinear equations in two and three variables.

The Method of Substitution

The Method of Substitution

Solving a System of Linear Equations in Two Variables Solve the system of equations. x + y = 4 x – y = 2 Solution: Solving for y in Equation 1. y = 4 – x Substitute y = 4 – x in Equation 2 and solve the resulting single-variable equation for x. Equation 1 Equation 2 Solve for y in Equation 1. Write Equation 2.

Example 1 – Solution x – (4 – x) = 2 x – 4 + x = 2 2x = 6 x = 3 cont’d x – (4 – x) = 2 x – 4 + x = 2 2x = 6 x = 3 Finally, you can solve for y by back-substituting x = 3 into the equation y = 4 – x, to obtain y = 4 – x y = 4 – 3 Substitute 4 – x for y. Distributive Property Combine like terms. Divide each side by 2. Write revised Equation 1. Substitute 3 for x.

Example 1 – Solution cont’d y = 1. The solution is the ordered pair (3, 1). You can check this solution as follows. Solve for y.

Solving a System of Linear Equations in Three Variables

Nonlinear Systems of Equations

Example 3 – Substitution: Two-Solution Case Solve the system of equations. 3x2 + 4x – y = 7 2x – y = –1 Solution: Solving for y in Equation 2 to obtain y = 2x + 1. Substitute y = 2x + 1 into Equation 1 and solve for x. 3x2 + 4x – (2x + 1) = 7 3x2 + 2x – 1 = 7 Equation 1 Equation 2 Substitute 2x + 1 for y in Equation 1. Simplify.

Example 3 – Solution 3x2 + 2x – 8 = 0 (3x – 4)(x + 2) = 0 cont’d 3x2 + 2x – 8 = 0 (3x – 4)(x + 2) = 0 Back-substituting these values of x to solve for the corresponding values of y produces the solutions and Write in general form. Factor. Solve for x.

Solve systems of nonlinear equations

What You Should Learn Use the method of elimination to solve systems of linear equations in two and three variables.

The Method of Elimination

The Method of Elimination

Solving a System of Linear Equations in Two Variables Solve the system of linear equations. Solution: For this system, you can obtain coefficients that differ only in sign by multiplying Equation 2 by 4. Equation 1 Equation 2 Write Equation 1. Multiply Equation 2 by 4. Add equations. Solve for x.

Example 2 – Solution cont’d By back-substituting into Equation 1, you can solve for y. 2x – 4y = –7 –4y = –7 The solution is Write Equation 1. Substitute for x. Combine like terms. Solve for y.

Solving a System of Linear Equations in Three Variables

What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities.

The Graph of an Inequality

The Graph of an Inequality

Example 1 – Sketching the Graph of an Inequality Sketch the graph of y  x2 – 1. Solution: Step 1: Change the inequality sign to an equal sign. y = x2 – 1 Step 2: Sketch the graph. Step 3: Test a point. Substitute (0,0) into y  x2 – 1 Substitute (0,-2) into y  x2 – 1 We can see that the points above (or on) the graph satisfy the inequality. Step 4: Shade the region containing the point (0,0)

Systems of Inequalities

Example 4 – Solving a System of Inequalities Sketch the graph (and label the vertices) of the solution set of the system. x – y < 2 x > –2 y  3 Inequality 1 Inequality 2 Inequality 3

Example 1 – Sketching the Graph of an Inequality Solution: Step 1: Change the inequality sign to an equal sign. y = x-2 x = –2 y = 3 Step 2: Sketch the graph. Step 3: Test a point for each inequality. Step 4: Shade the region. Step 5: The overlapping region is the solution.

Example 4 – Solution To find the vertices of the region, solve: cont’d To find the vertices of the region, solve: Vertex A: x – y = 2 x = –2 A(–2, –4) Vertex B: x – y = 2 y = 3 B(5, 3) Vertex C: x = –2 y = 3 C(–2, 3)