ALGEBRA 2: CHAPTER Solve Linear Systems by Graphing
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1 ALGEBRA 2: CHAPTER Solve Linear Systems by Graphing Goal Solve systems of linear equations. Your Notes VOCABULARY System of two linear equations Two equations, with the variables x and y that can be written as: Ax + By = C Equation 1 Dx + Ey = F Equation 2 Solution of a system An ordered pair (x, y) that satisfies each equation Consistent A system that has at least one solution Inconsistent A system that has no solution Independent A consistent system that has exactly one solution Dependent A consistent system that has infinitely many solutions Example 1 Solve a system graphically Graph the system and estimate the solution. Then check the solution algebraically. 4x + 2y = 4 Equation 1 2x 3y = 10 Equation 2
2 Checkpoint Graph the linear system and estimate the solution. Then check the solution algebraically. 1. 4x + y = 2 6x 3y = 12 NUMBER OF SOLUTIONS OF A LINEAR SYSTEM Exactly one solution Infinitely many solutions No Solutions Lines intersect at one point consistent and independent Infinitely many solutions it y Lines coincide; consistent and dependent Lines are _parallel_; _inconsistent_ Example 2 Solve a system with many solutions Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent. 2x + y = 4 Equation 1 4x 2y = 8 Equation 2
3 Example 3 Solve a system with no solution Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent. 2x + 4y = 8 Equation 1 2x + 4y = 4 Equation 2 Checkpoint Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent. 2. 3x 2y = 6 5x + 4y = 8 3. x 2y = 5 2x 4y = x 3y = 12 6x 3y = 6
4 5. x + y = 2 4x 3y = 1 Example 4 Writing and using a linear system Ice Cream Shop At an ice cream shop, one customer pays $9 for 2 sundaes and 2 milkshakes. A second customer pays $13 for 2 sundaes and 4 milkshakes. How much do each sundae and milkshake cost? Checkpoint Complete the following exercise. 6. In Example 4, how much do each sundae and milkshake cost if the first customer pays $7 and the second customer pays $10?
5 3.2 Solve Linear Systems Algebraically Goal Solve linear systems algebraically. Your Notes VOCABULARY Substitution method Substitute an expression into one of the equations to solve for the variable Elimination method Eliminate one of the variables by adding equations THE SUBSTITUTION METHOD Step 1 Solve one of the equations for one of its variables. Step 2 Substitute the expression from _Step 1_ into the other equation and solve for the other variable. Step 3 Substitute the value from _Step 2_ into the revised equation from Step 1 and solve. Example 1 Use the substitution method Solve the system using the substitution method. x + 2y = 2 Equation 1 3x + 4y = 6 Equation 2 THE ELIMINATION METHOD Step 1 Multiply one or both of the equations by a _constant_ to obtain coefficients that differ only in _sign_ for one of its variables. Step 2 Add the revised equations from _Step 1 _.Combining like terms will _eliminate one of the variables. Solve for the remaining variable. Step 3 Substitute the value obtained in _Step 2 _into either of the original equations and solve for the other variable.
6 Example 2 Use the elimination method Solve the system using the elimination method. 2x + 5y = 14 Equation 1 4x + 2y = 4 Equation 2 Checkpoint Complete the following exercises. 1. Solve the linear system using the substitution method. 2x + y = 2 5x + 3y = 8 2. Solve the linear system using the elimination method. 3x + 8y = 5 2x + 2y = 18
7 Example 3 Use the elimination method Solve the linear system using the elimination method. 3x 4y = 37 Equation 1 5x + 3y = 14 Equation 2 Example 4 Solve linear systems with many or no solutions Solve the linear system. a. x 3y = 7 b. 2x 6y = 12 2x 6y = 12 5x + 15y = 30 Checkpoint Solve the linear system. 3. 2x y = x + 3y = 7 8x 4y = 13 5x + 7y = 15
8 3.3 Graph Systems of Linear Inequalities Goal Graph systems of linear inequalities Your Notes VOCABULARY System of linear inequalities A system of two or more linear inequalities in two variables Solution of a system of linear inequalities An ordered pair that is a solution of each inequality in the system Graph of a system of linear inequalities The graph of all solutions of the system GRAPHING A SYSTEM OF LINEAR INEQUALITIES To graph a system of linear inequalities, follow these steps: Step 1 Graph each inequality in the system. You may want to use colored pencils to distinguish the different _half-planes_. Step 2 Identify the region that is _common_ to all the graphs of the inequalities. This region is the graph of the system. If you used colored pencils, the graph of the system is the region that has been shaded with _every_color. Example 1 Graph a system of two inequalities Graph the system. y 2x + 1 Inequality 1 y x 3 Inequality 2
9 Example 2 Graph a system with no solution Graph the system. 3 y x+ 1 4 Inequality 1 3x + 4y 16 Inequality 2 Checkpoint Graph the system of inequalities. 1. y 2x 3 1 y x x + 3y 6 1 y x + 1 3
10 Example 3 Graph a system with an absolute value inequality Graph the system. y 4 Inequality 1 y x Inequality 2 Checkpoint Graph the system. 3. y x 1 1 y x 2 2 x 2
11 3.4 Solve Systems of Linear Equations in Three Variables Goal Solve systems of equations in three variables. VOCABULARY Linear equation in three variables An equation of the form ax+ by+ cz.= d, where a, b, and c are not all zero System of three linear equations A system made up of three linear equations in three variables Solution of a system of three linear equations The solution is the values of the three variables that make each equation true. Ordered triple A coordinate in three variables (x, y, z) THE ELIMINATION METHOD FOR A THREE-VARIABLE SYSTEM Step 1 Rewrite the linear system in three variables as a linear system in two variables by using the elimination method. Step 2 Solve the new linear system for both of its variables. Step 3 Substitute the values found in step 2 into one of the original equations and solve for the remaining variable. If you obtain a _false_ equation, such as 0 = 1, in any of the steps, then the system has no _solution._ If you do not obtain a false equation, but obtain an _identity_ such as 0 = 0, then the system has _infinitely many solutions_.
12 Example 1 Use the elimination method Solve the system. 3x 2y + 4z = 20 Equation 1 x + 5y + 12z = 73 Equation 2 x + 3y 2z = 1 Equation 3
13 Checkpoint Solve the linear system. 1. 4x 3y + 5z = 19 3x + y 8z = 21 2x + y + 3z = 13
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