Chapter 4. Section 4.1: Solving Systems of Linear Equations by Graphing


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1 Chapter 4 Section 4.1: Solving Systems of Linear Equations by Graphing Objectives: 1. Decide whether a given ordered pair is a solution of a system. 2. Solve linear systems by graphing. 3. Solve special systems by graphing. 4. Identify special systems without graphing. Deciding Whether a Given Ordered Pair is a Solution A system of linear equations, often called a linear system, consists of two or more linear equations with the same variables. A solution of a system of linear equations is an ordered pair that makes both equations true at the same time. A solution of an equation is said to satisfy the equation. Example: Is (2, 1) a solution of the system 3x + y = 5 2x 3y = 7? Example: Is (2, 1) a solution of the system x + 5y = 3 4x + 2y = 1? 1 P a g e
2 Solving Linear Systems by Graphing Example 2 Solve the system of equations by graphing both equations on the same axes. 2x + ⅔y = 4 5x y = 8 Step1: Rewrite each equation in slopeintercept form to graph. Step 2: Graph both lines on the same axes and identify where they cross. CAUTION A difficulty with the graphing method of solution is that it may not be possible to determine from the graph the exact coordinates of the point that represents the solution, particularly if these coordinates are not integers. For this reason, algebraic methods of solution are explained later in this chapter. The graphing method does, however, show geometrically how solutions are found and is useful when approximate answers will do. 2 P a g e
3 Solving Special Systems by Graphing Solve each system by graphing. (a) 3x + y = 4 6x + 2y = 1 3 P a g e
4 (b) ½x + y = 3 2x + 4y = 12 4 P a g e
5 Three Cases for Solutions of Systems 1. The graphs intersect at exactly one point, which gives the (single) orderedpair solution of the system. The system is consistent, and the equations are independent. 2. The graphs are parallel lines. So, there is no solution and the solution set is Ø. The system is inconsistent and the equations are independent. 3. The graphs are the same line. There is an infinite number of solutions, and the solution set is written in setbuilder notation. The system is consistent and the equations are dependent. 5 P a g e
6 Section 4.2: Solving Systems of Linear Equations by Substitution Objectives: 1. Solve linear systems by substitution. 2. Solve special systems by substitution. 3. Solve linear systems with fractions and decimals by substitution. Solving Linear Systems by Substitution Step 1 Solve one equation for either variable. If one of the variables has a coefficient of 1 or 1, choose it, since it usually makes the substitution easier. Step 2 Substitute for that variable in the other equation. The result should be an equation with just one variable. Step 3 Solve the equation from Step 2. Step 4 Substitute the result from Step 3 into the equation from Step 1 to find the value of the other variable. Step 5 Check the solution in both of the original equations. Then write the solution set. Example: Solve the system by the substitution method. The second equation is already solved for y. This equation says that y = 3x. Substituting 3x for y in the first equation gives Because x = 2, we find y from the equation y = 3x by substituting 2 for x. Check that the solution of the given system is ( 2, 6) by substituting 2 for x and 6 for y in both equations. The solution set of the system is {( 2, 6)}. 6 P a g e
7 Example: Solve the system by the substitution method. 7 P a g e
8 Example: Solve the system by the substitution method. 8 P a g e
9 Example: Solve the system by the substitution method. 9 P a g e
10 Example: Solve the system by the substitution method. 10 P a g e
11 Section 4.3: Solving Systems of Linear Equations by Elimination Objectives: 1. Solve linear systems by elimination. 2. Multiply when using the elimination method. 3. Use an alternative method to find the second value in a solution. 4. Use the elimination method to solve special systems. Solving Linear Systems by Elimination Using the addition property to solve systems is called the elimination method. When using this method, the idea is to eliminate one of the variables. To do this, one of the variables in the two equations must have coefficients that are opposites. Solving a Linear System by Elimination Step 1 Write both equations in standard form Ax + By = C. Step 2 Transform so that the coefficients of one pair of variable terms are opposites. Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either the x or yterms is 0. Step 3 Add the new equations to eliminate a variable. The sum should be an equation with just one variable. Step 4 Solve the equation from Step 3 for the remaining variable. Step 5 Substitute the result from Step 4 into either of the original equations and solve for the other variable. Step 6 Check the solution in both of the original equations. Then write the solution set. Example: Use the elimination method to solve the system. 11 P a g e
12 CAUTION A system is not completely solved until values for both x and y are found. Do not stop after finding the value of only one variable. Remember to write the solution set as a set containing an ordered pair. Example: Use the elimination method to solve the system. 12 P a g e
13 Example: Use the elimination method to solve the system. 13 P a g e
14 Example: Use the elimination method to solve the system. 14 P a g e
15 Section 4.4: Applications of Linear Systems Objectives: 1. Solve problems about unknown numbers. 2. Solve problems about quantities and their costs. 3. Solve problems about mixtures. 4. Solve problems about distance, rate (or speed), and time. Solving an Applied Problem with Two Variables Step 1 Read the problem, several times if necessary, until you understand what is given and what is to be found. Step 2 Assign variables to represent the unknown values, using diagrams or tables as needed. Write down what each variable represents. Step 3 Write two equations using both variables. Step 4 Solve the system of two equations. Step 5 State the answer to the problem. Is the answer reasonable? Step 6 Check the answer in the words of the original problem. Example: Amy and Bryan just celebrated a birthday together. Amy is 3 years older than Bryan. However, if Bryan doubled his age and subtracted 5, the result would be 4 years older than Amy. How old are Amy and Bryan? 15 P a g e
16 Example: Carrie and Diego are headed to the vending machine and have decided to pool their money. Carrie has only quarters, Diego has only dimes, and together they have ten coins worth $2.05. How many quarters does Carrie have, and how many dimes does Diego have? 16 P a g e
17 Example: A pharmacist needs 100L of 50% alcohol solution. She has on hand 30% alcohol solution and 80% alcohol solution, which she can mix. How many liters of each will be required to make the 100L of 50% alcohol solution? 17 P a g e
18 Example: A plane flies 560 mi in 1.75 hr traveling with the wind. The return trip against the same wind takes the plane 2hr. Find the speed of the plane and the speed of the wind. 18 P a g e
19 Section 4.5: Solving Systems of Linear Inequalities Objectives: 1. Solve systems of linear inequalities by graphing. Solving a System of Linear Inequalities Step 1 Graph the inequalities. Graph each inequality using the method of Section 3.5. Step 2 Choose the intersection. Indicate the solution set of the system by shading the intersection of the graphs (the region where the graphs overlap). Example: Graph the solution set of the system. 19 P a g e
20 Example: Graph the solution set of the system. 20 P a g e
21 Example: Graph the solution set of the system. 21 P a g e
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