TMAT 103Chapter 6Systems of Linear Equations

TMAT 1036.1Solving a System of Two Linear Equations

6.1 Solving a System of Two Linear EquationsSystems of two linear equations:

6.1 Solving a System of Two Linear EquationsGraphs of linear systems of equations with two variablesThe two lines may intersect at a common, single point. This point, in ordered pair form(x, y), is the solution of the systemindependent and consistentThe two lines may be parallel with no points in common; hence, the system has no solutioninconsistentThe two lines may coincide; the solution of the system is the set of all points on the common linedependent

6.1 Solving a System of Two Linear EquationsMethods available to solve systems of equationsAddition-subtraction methodMethod of substitution

6.1 Solving a System of Two Linear EquationsSolving a pair of linear equations by the addition-subtraction methodIf necessary, multiply each side of one or both equations by some number so that the numerical coefficients of one of the variables are of equal absolute value.If these coefficients of equal absolute value have like signs, subtract one equation from the other. If they have unlike signs, add the equations.Solve the resulting equation for the remaining variable.Substitute the solution for the variable found in step 3 in either of the original equations, and solve this equation for the second variable.Check

6.1 Solving a System of Two Linear EquationsExamples solve the following using the addition-subtraction method

6.1 Solving a System of Two Linear EquationsSolving a pair of linear equations by the method of substitutionFrom either of the two given equations, solve for one variable in terms of the other.Substitute this result from step 1 in the other equation. Note that this step eliminates one variable.Solve the equation obtained from step 2 for the remaining variable.From the equation obtained in step 1, substitute the solution for the variable found in step 3, and solve this resulting equation for the second variable.Check

6.1 Solving a System of Two Linear EquationsExamples solve the following using the method of substitution

6.1 Solving a System of Two Linear EquationsSteps for problem solvingRead the problem carefully at least two times.If possible, draw a picture or diagram.Write what facts are given and what unknown quantities are to be found.Choose a symbol to represent each quantity to be found.Write appropriate equations relating these variables from the information given in the problem (there should be one equation for each unknown).Solve for the unknown variables using an appropriate method.Check your solution in the original equation.Check your solution in the original verbal problem.

6.1 Solving a System of Two Linear EquationsExamples solve the following

A plane can travel 900 mile with the wind in 3 hours. It makes the return trip in 3.5 hours. Find the rate of windspeed, and the speed of the plane

A chemist has a 5% solution and an 11% solution of acid. How much of each must be mixed to get 1000L of a 7% solution?

TMAT 1036.2Other systems of equations

6.2 Other systems of equationsOther types of problems can be solved using either the addition-subtraction method, or the method of substitutionLiteral equationscoefficients are letterswill not be covered in this classNon-linear equationsvariables in denominator

Examples solve the following using the method of substitution or addition-subtraction method

6.2 Other systems of equations

TMAT 1036.3Solving a System of Three Linear Equations

6.3 Solving a System of Three Linear EquationsSystems of three linear equations:

6.3 Solving a System of Three Linear EquationsGraphs of linear systems of equations with three variablesThe three planes may intersect at a common, single point. This point, in ordered triple form (x, y, z), is then the solution of the system.The three planes may intersect along a common line. The infinite set of points that satisfy the equation of the line is the solution of the system.The three planes may not have any points in common; the system has no solution. For example, the planes may be parallel, or they may intersect triangularly with no points in common to all three planes.The three planes may coincide; the solution of the system is the set of all points in the common plane.

6.3 Solving a System of Three Linear EquationsSolving a pair of linear equations by the addition-subtraction methodEliminate a variable from any pair of equations using the same technique from section 6.1Eliminate the same variable from any other pair of equations.The results of steps 1 and 2 is a pair of linear equations in two unknowns. Solve this pair for the two variablesSolve for the third variable by substituting the results from step 3 in any one of the original equationsCheck

6.3 Solving a System of Three Linear EquationsExamples solve the following using the addition-subtraction method

6.3 Solving a System of Three Linear EquationsExample solve the following

75 acres of land were purchased for $142,500. The land facing the highway cost $2700/acre. The land facing the railroad cost $2200/acre, and the remainder cost $1450/acre. There were 5 acres more facing the railroad than the highway. How much land was sold at each price?