Chapter Seven 7.2 – Systems of Linear Equations in Two Variables

7.2 – Systems of Linear Equations in Two Variables The Method of Elimination Graphical Interpretation of Two-Variable Systems Applications

7.2 – The Method of Elimination 1.Create opposite coefficients on one variable. 2.Add the equations to eliminate one variable; solve the resulting equation. 3.Back-substitute into the first equation. 4.Check your solutions in both original equations.

Example 1 Solve the system of linear equations. 5x + 3y = 9 2x - 4y = 14 (3, -2)

Example 1 - You try Solve the system of linear equations by elimination. 4x + 3y = 3 3x + 11y = 13

7.2 – Graphical Interpretation of Two- Variable Systems There three possibilities for linear systems: 1.One solution 2.Infinitely many solutions (coinciding lines) Equations with at least one solution are called consistent. 3.No solution (parallel lines) These equations are inconsistent.

If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. Solution y x

If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. x y

If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent. x y

Example 2 Solve the system of linear equations. x - 2y = 3 -2x + 4y = 1

7.2 Applications Find the value of k, such that the system is inconsistent. k= 2

Example 3 - You try Solve the system of linear equations. 2x - y = 1 4x - 2y = 2

Example 4 - You try Solve the system of linear equations.

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7.2 Applications  Point of Equilibrium: Point where supply and demand equations are equivalent. Find the equilibrium point for the supply and demand equations: Demand: x Supply: x

 Distance, rate, time problems: True rate incorporates current or wind. You row 8 miles downstream (with the current) in 1 hour and 6 miles upstream (against the current) in 3 hours. How fast can you row in still water? What is the rate of the current?

Combining rates: An airplane flying into a headwind travels the 2000 mile flying distance between Fresno, California and Cleveland, Ohio in 4 hours and 24 minutes. On the return flight, the same distance is traveled in 4 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant. Plane: mph Wind: mph

7.2 Applications A nurse wishes to obtain 800 ml of a 7% solution of boric acid by mixing 4% and 12% solutions. How much of each solution should be used? 300 ml of the 12% solution and 500 ml of the 4%

7.2 Applications You have $10,000 invested in 2 funds. One pays 8% simple interest, the other pays 6% simple interest. How much did you invest in each fund if you earned total interest of $720 in one year? $4000 in the 6% fund and $6000 in the 8% fund.

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