Chapter 4Systems of Linear Equations; MatricesSection 1Review: Systems of Linear Equations in Two yqVariablesOpening ExampleA restaurant serves two types of fish dinners-small for $5 99A restaurant serves two types of fish dinnerssmall for $5.99 and large for $8.99. One day, there were 134 total orders of fish, and the total receipts for these 134 orders was $1,024.66. How many small fish dinners and how many large fish dinners were ordered? 2Systems of Two Equations in Two VariablesWe are given the linear systemWe are given the linear system ax + by = h cx + dy = k A solution is an ordered pair (x0, y0) that will satisfy each ti ( k t ti h b tit t d i t th t3equation (make a true equation when substituted into that equation). The solution set is the set of all ordered pairs that satisfy both equations. In this section, we wish to find the solution set of a system of linear equations. To solve a system is to find its solution set. Solve by Graphing One method to find the solution of a system of linearOne method to find the solution of a system of linear equations is to graph each equation on a coordinate plane and to determine the point of intersection (if it exists). The drawback of this method is that it is not very accurate in most cases, but does give a general location of the point of intersection. Lets take a look at an example: Solve the following system by graphing:4Solve the following system by graphing: 3x + 5y = –9 x+ 4y = –10Solve by GraphingSolution3x + 5y = –9 +410x+ 4y= –10 First line (intercept method)If x = 0, y= –9/5 If y = 0 , x = –3 Plot points and draw line(2, –3)Second line:5Second line: Intercepts are (0, –5/2) , ( –10,0)From the graph we estimate that the point of intersection is (2,–3). Check: 3(2) + 5(–3)= –9 and 2 + 4(–3)= –10. Both check.Another ExampleNow you try one:Now, you try one:  Solve the system by graphing: 2x + 3 = yx + 2y = –4 6Another ExampleSolutionNow you try one:Now, you try one:  Solve the system by graphing: 2x + 3 = yx + 2y = –4 We can do this on a graphing calculator by first solving the 7second equation for yto get y= –0.5x–2If we enter both of these equations into the calculator and find the intersection point we get the screen shown.The solution is (–2,–1).Method of Substitution Although the method of graphing is intuitive it is not veryAlthough the method of graphing is intuitive, it is not very accurate in most cases, unless done by calculator. There is another method that is 100% accurate. It is called the method of substitution. This method is an algebraic one. It works well when the coefficients of x or y are either 1 or –1. For example, let’s solve the previous system2x+3=y82x+ 3 yx + 2y = -4 using the method of substitution.Method of Substitution(continued)The steps for this method are as follows:The steps for this method are as follows:1) Solve one of the equations for either x or y.2) Substitute that result into the other equation to obtain an equation in a single variable (either x or y). 3) Solve the equation for that variable. 4) Substitute this value into any convenient equation to 9obtain the value of the remaining variable.Example(continued)2x+3=ySubstitute yfrom first 2x + 3 yx + 2y = –4 x + 2(2x + 3) = –4x + 4x + 6 = –45x + 6 = –4 Substitute yfrom first equation into second equation Solve the resulting equation After we find x = –2, then 105x = –10 x = –2 from the first equation, we have 2(–2) + 3 = y or y = –1. Our solution is (–2, –1) Another Example Solve the system usingSolve the system using substitution: 3x –2y = 7y = 2x –311Another Example SolutionSolve the system usingSolution:Solve the system using substitution: 3x –2y = 7y = 2x –3Solution:32 72332(23)73467xyyxxxxx−==−−−=−+=12()11213 5xxyy−==−=−−→=−Basic Terms  A consistent linear system is one that has one or more solutionssolutions. • If a consistent system has exactly one solution (unique solution), it is said to be independent. An independent system will occur when the two lines have different slopes. • If a consistent system has more than one solution, it is said to be dependent. A dependent system will occur when the t li h th l d thit tI13two lines have the same slope and the same yintercept. In other words, the graphs of the lines will coincide. There will be an infinite number of points of intersection. Two systems of equations are equivalent if they have the same solution set.Basic TermsAninconsistentlinear system is one that has no solutionsAn inconsistentlinear system is one that has no solutions. This will occur when two lines have the same slope but different y intercepts. In this case, the lines will be parallel and will never intersect.  Example: Determine if the system is consistent, independent, dependent or inconsistent: 2x5y=6142x–5y= 6 –4x + 10y = –1 Solution of Example Solve each equation for yto obtain the slope intercept form of the qyppequation: 25 62652652655xyxyxyxy−=−=−=−=410 110 4 1412110 10 5 10xyyxxyx−+=−=−=−= −15 Since each equation has the same slope but different y intercepts, they will not intersect. This is an inconsistent system.55Elimination by Addition The method of substitution is not preferable if none of the coefficients ofxandyare 1 or1 For example substitution iscoefficients of xand yare 1 or –1. For example, substitution is not the preferred method for the system below: 2x –7y = 3–5x + 3y = 7  A better method is elimination by addition. The following operations can be used to produce equivalent systems: 16• Two equations can be interchanged. • An equation can be multiplied by a non-zero constant. • A constant multiple of one equation and is added to another equation.Elimination by AdditionExample For our system, we will seek to eliminate the x variable. The 2x − 7y = 3 −5x + 3y = 7⎫⎬⎭→coefficients are 2 and –5. Our goal is to obtain coefficients of x that are additive inverses of each other. We can accomplish this by multiplying the first equation by 5, and the second equation by 2. Next, we can add the two equations {⎭5(2x − 7 y) = 5(3)2(−5x + 3y) = 2(7)⎫⎬⎭→10x − 35y = 15−10x + 6y = 14⎫⎬⎭→0x − 29y = 29 →117to eliminate the x-variable.  Solve for y Substitute y value into original equation and solve for x Write solution as an ordered