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TMAT 103

TMAT 103. Chapter 6 Systems of Linear Equations. TMAT 103. § 6.1 Solving a System of Two Linear Equations. §6 .1 – Solving a System of Two Linear Equations. Systems of two linear equations:. §6 .1 – Solving a System of Two Linear Equations.

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TMAT 103

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  1. TMAT 103 Chapter 6 Systems of Linear Equations

  2. TMAT 103 §6.1 Solving a System of Two Linear Equations

  3. §6.1 – Solving a System of Two Linear Equations • Systems of two linear equations:

  4. §6.1 – Solving a System of Two Linear Equations • Graphs of linear systems of equations with two variables • The two lines may intersect at a common, single point. This point, in ordered pair form(x, y), is the solution of the system • independent and consistent • The two lines may be parallel with no points in common; hence, the system has no solution • inconsistent • The two lines may coincide; the solution of the system is the set of all points on the common line • dependent

  5. §6.1 – Solving a System of Two Linear Equations • Methods available to solve systems of equations • Addition-subtraction method • Method of substitution

  6. §6.1 – Solving a System of Two Linear Equations • Solving a pair of linear equations by the addition-subtraction method • If 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

  7. §6.1 – Solving a System of Two Linear Equations • Examples – solve the following using the addition-subtraction method

  8. §6.1 – Solving a System of Two Linear Equations • Solving a pair of linear equations by the method of substitution • From 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

  9. §6.1 – Solving a System of Two Linear Equations • Examples – solve the following using the method of substitution

  10. §6.1 – Solving a System of Two Linear Equations • Steps for problem solving • Read 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.

  11. §6.1 – Solving a System of Two Linear Equations • Examples – 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?

  12. TMAT 103 §6.2 Other systems of equations

  13. §6.2 – Other systems of equations • Other types of problems can be solved using either the addition-subtraction method, or the method of substitution • Literal equations • coefficients are letters • will not be covered in this class • Non-linear equations • variables in denominator

  14. §6.2 – Other systems of equations • Examples – solve the following using the method of substitution or addition-subtraction method

  15. TMAT 103 §6.3 Solving a System of Three Linear Equations

  16. §6.3 – Solving a System of Three Linear Equations • Systems of three linear equations:

  17. §6.3 – Solving a System of Three Linear Equations • Graphs of linear systems of equations with three variables • The 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.

  18. §6.3 – Solving a System of Three Linear Equations • Solving a pair of linear equations by the addition-subtraction method • Eliminate a variable from any pair of equations using the same technique from section 6.1 • Eliminate 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 variables • Solve for the third variable by substituting the results from step 3 in any one of the original equations • Check

  19. §6.3 – Solving a System of Three Linear Equations • Examples – solve the following using the addition-subtraction method

  20. §6.3 – Solving a System of Three Linear Equations • Example – 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?

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