8.1 Solving Systems of Equations Graphically

8.1 Solving Systems of Equations Graphically Definitions System of Equations involves equations that contain the same variables In this section we will look at both linear-quadratic systems and quadratic-quadratic systems of equations. The solution to a system of equation are the ordered pairs (x, y) that satisfy both equations. Graphically, the solution is represented by the point(s) of intersection of the equations. There are several ways to solve systems of equations Graphically Algebraically (Substitution or Elimination) Example 1: Solve the following linear-linear system graphically a) b) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 1 of 40

Example 2: Solve the following system of equations a) b) c) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 2 of 40

d) e) Example 3: Determine the potential number of solutions in a: a) Linear-quadratic system b) Quadratic-quadratic system Assignment Page 435, #4abd, 8, Worksheet #9 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 3 of 40

8.1 Worksheet 9. Solve the following systems graphically. a) b) c) d) e) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 4 of 40

8.1 Worksheet Solutions Answers: 9 a) (4, 7), (2, 3) b) No Solution c) ( 2, 0), (0, 2) d) (4, 3), (5, 2) e) ( 2, 7) (2, 1) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 5 of 40

8.2 Solving Systems of Equations Algebraically The solution to a system of equation are the ordered pair(s (x, y) that satisfy both equations. Graphically, the solution is represented by the point(s) of intersection of the equations. There are several ways to solve systems of equations. Graphically Algebraically (Substitution or Elimination) To solve a system of equations using an algebraic approach, there are two methods we can use: Substitution: In this method you will isolate either variable in one equation and substitute that expression into the other equation and solve. Once you have solved for one variable, you will substitute it back into the expression to get the other. Elimination: In this method, you will rearrange the terms in the each of the equations so all like terms are aligned. You will multiply all the terms of the equations by a non-zero number so that the coefficients for one of the variables is the same. Add or subtract the equations together so that one of the variables is eliminated. Solve for the variable then substitute the value into either equation to get the value of the other. You must decide which method to use when solving a system of equations. Certain systems are easier to solve with one method over the other. Example 1a: Solve the following system of equations using substitution. x y 8 3x 2y 14 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 6 of 40

Example 1b: Solve the following system of equations using elimination: x y 3x 2y 8 14 Example 2: Solve the system of equations. Which method would be easier? y 2x 3x 4 y 11 Example 3a: Solve the system of equations using substitution 3x y 4x 2 x y 9 9 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 7 of 40

Example 3b: Solve the system of equations using elimination 3x y 4x 2 x y 9 9 Example 4: Solve the system of equations. 2x 2 4x y 3 4x 2y 7 Example 5: Solve the system of equations. 2 x y 8x 19 2 x y 7x 11 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 8 of 40

Example 6: Solve the system of equations. 6x 2 x y 4x 2 4x y 1 6 Assignment Page 451, #3-5, 9, 10 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 9 of 40

Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 10 of 40

9.2 Quadratic Inequalities in One Variable Investigate: Given the graph of y 2 x 3x 4 below, we want to examine the y-values for different parts of the function. On the graph of the function, where are the y-values positives? What are the x-values that give these positive y-values? 2 Solve x 3x 4 0 2 Given the inequality x 3x 4 0, we are solving the exact situation above. We are finding the x- values of the function y 2 x 3x 4 that will yield positive y-values on the curve. Quadratic Inequalities in One Variable A quadratic inequality in one variable can be written in standard form as: 2 Example 1: Given the graph of y x 3x 4 below, 2 Solve x 3x 4 0 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 11 of 40

Example 2: Given the graph of y f (x) below, a) Solve f (x) 0 b) Solve f (x) 0 Example 3: Given the graph of y f (x) below, a) Solve f (x) 0 b) Solve f (x) 0 Example 4: Given the graph of y f (x) below, a) Solve f (x) 0 b) Solve f (x) 0 c) Solve f (x) 0 d) Solve f (x) 0 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 12 of 40

If the graph of the function is not given to you, then you must draw a sketch of the parabola. When drawing a sketch of the curve there are two important pieces of information that are required: 1. Zeros of the function 2. Direction of opening of the parabola. You must bring all the terms to one side of the inequality and factor the quadratic expression to find the zeros of the function. If the expression is not factorable then you must apply the quadratic formula to find the values of x. Example 5: Solve the following inequalities. a) Solve ( x 1)( x 3) 0 b) Solve 2( x 2)( x 1) 0 2 c) Solve x 2x 3 0 d) Solve Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 13 of 40

e) Solve f) Solve g) Solve Assignment Page 484, #1, 2, 3ab, 4-9 (ac) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 14 of 40

9.1 Linear Inequalities in Two Variables Linear Inequality in Two Variables A linear inequality in two variables can be expressed in the following forms A linear inequality in two variables describes a region of points on the Cartesian coordinate plane. Any ordered pair (x, y) that satisfies the inequality is a part of the solution region. The linear function is the boundary that divides the plane into two regions. For one region For one region is true. is true. After graphing the function, you can use a test point to determine which region to shade. The test point is point not on the boundary of the graph and is substituted into the inequality to test whether the inequality holds true or not. When graphing the inequality the form : Find and plot x and y intercepts of the corresponding equation. If the equality involves > or <, connect the points with a dashed line. If the equality involves or, connect the points with a solid line. Substitute the test point into the inequality. o If the point satisfies the inequality shade on that side of the boundary that contained the test point. o If the point does not satisfies the inequality shade on the opposite side of the boundary that contained the test point. Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 16 of 40

Example 1: Graph each inequality a) b) If the x and y intercepts are not integer values, it is difficult to precisely graph the function. Another way to graph is to re-write the function into form.** Linear Inequality in Two Variables A linear inequality in two variables can be expressed in the following forms When graphing the function in the form : Plot the y-intercept (b) and find additional points on the line using the slope (m) If the equality involves > or <, connect the points with a dashed line. If the equality involves or, connect the points with a solid line. If the inequality involves > or, shade above the boundary line. If the inequality involves < or, shade below the boundary line. ** When re-writing an inequality into slope-intercept form, remember that if you multiply or divide both sides of the inequality by a negative value, you must reverse the inequality sign. Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 17 of 40

Example 2: Graph the following inequalities a) b) c) d) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 18 of 40

Example 3: Write an inequality to represent each graph a) b) Example 4: Graph each inequality a) b) Assignment Page 472, #1ad, 2ac, 3bdf, 4, 9 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 19 of 40

9.3 Quadratic Inequalities in Two Variables Last class we discovered how to graph linear inequalities in two variables. We will use those same principles to graph quadratic inequalities in two variables. When graphing the function in the form : Plot points on the corresponding equation If the equality involves > or <, connect the points with a dashed line. If the equality involves or, connect the points with a solid line. If the inequality involves > or, shade above the boundary line. If the inequality involves < or, shade below the boundary line. Example 1: Graph the following inequalities: a) b) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 21 of 40

c) d) e) f) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 22 of 40

g) h) Assignment Page 496, #1ab, 2bc, 3, 4-6, 8 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 23 of 40

9.4* Solving Systems of Inequalities in two variables. A system of inequalities in two variables involves inequalities that contain the same variables, The solution to a system of inequalities in two variables is the set of ordered pairs that satisfies all the inequalities in the system. The solution is found by shading the overlapping regions for all inequalities. Example1: Solve the system of inequalities by graphing. a) b) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 25 of 40

c) d) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 26 of 40

Ch. 8 & 9 Review Key Ideas Solving systems of Equations (Linear-Quadratic, Quadratic-Quadratic) o Graphically o Algebraically Substitution Elimination Solving quadratic inequalities in one variable. Graphing linear inequalities in two variables. Graphing quadratic inequalities in two variables. 1. Solve the following system of equations a) b) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 28 of 40

2. Solve the following system of equations. a) b) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 29 of 40

c) 3. Solve the following inequalities a) b) c) d) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 30 of 40

4. Graph the following inequalities. a) b) c) Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 31 of 40

d) Assignment Page 457, # 2, 3, 10 Page 459, # 7, 9 Page 501, # 1, 2, 6, 7ab, 11, 12cd, 13 Ch. 8 & 9: Systems of Equations / Linear & Quadratic Inequalities Lee/Ko 32 of 40