Math 11 Principles Chapter 5 Review

Some definitions: A system of equations consists of two or more equations that are considered together. A linear system of equations is a system in which each equation represents a straight line. A nonlinear system of equations is a system in which at least one equation does not represent a straight line. The solution to a system of equations must satisfy each equation in the system.

Example 1: Solve the system of equations graphically (you can also use your Graphing Calc) 2x + y = 2 x – 3y = 15

Solving linear systems by graphing is time-consuming and does not always give exact solutions. Therefore, we need to develop algebraic ways to solve them. They all involve eliminating one of the variables in some way. With the elimination (addition/subtraction) method, keep the following properties of linear systems in mind:  adding or subtracting equations does not change a solution  multiplying both sides of an equation will not change a solution (use your graphing calculator to check this out!)

Example 2: Solve the following by addition/subtraction. 4x + 3y = 3 3x – 2y = -19 There is also the method of SUBSTITUTION to solve these system of equations. To solve,  choose the simpler equation and express one variable in terms of the other  substitute the expression from above to the other equation  solve the single variable equation  substitute the solution into the 1st equation to find the value of the other variable

Example 3: Solve the following system using the substitution method. x + 3y = 11 2x – 3y = 4

Example 4: Solve by any method. a) y  x 2  4 b) xy  12 y  x  5 2x  y  2

You should be able to also solve word problems.

Example 5: Jane has $4.80 in nickels and quarters. She has 6 more nickels than quarters. How many of each does she have? Example 6: You can buy 6 cheap golf balls and 4 expensive ones for $12.50. You can buy 4 cheap and 3 expensive balls for $9.00. What are the prices of two kinds of balls?

You can use the same principles of the addition/subtraction method to solve a linear system with three variables. Our goal is to try to reduce/eliminate one of the variables to get the system to become a two equation system, then solve it just like before.

Example 7: Solve 2x  3y  z  12  2x  4y  3z  13 x  2y  3z  5 Linear Inequalities in Two Variables

The following are examples of linear inequalities of two variables. 2x  3y  4, 6x  y  1

The solution region to a linear inequality in two variables can be represented on a coordinate plane using a boundary line.

The boundary line will be solid or broken according to…  a solid line if the inequality is  or   a broken line if the inequality is < or >

To graph a linear inequality…  on a coordinate plane, graph the corresponding linear equation using a table of values, or intercepts, or the slope-intercept form. Make the line solid or broken depending on the sign  the solution region will be on one side of the line. To choose which side, use a test point (make sure the point is not on the line) and determine if the coordinates of the test point satisfy the inequality.  If the inequality is satisfied, then the solution is the region from which the test point was chosen. If not, it’s the other side.

Example 8: Graph the following inequalities. a) 2x  3y  12 b) x  y  6 2x  y  4 x  0