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8.1 Systems of Equations

8.1 Systems of Equations. A solution of a system of equations in two variables is any ordered pair that satisfies all equations in the system If in the coordinate plane, it is a point common to all graphs Linear system : system comprised entirely of linear equations

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8.1 Systems of Equations

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  1. 8.1 Systems of Equations

  2. A solution of a system of equations in two variables is any ordered pair that satisfies all equations in the system If in the coordinate plane, it is a point common to all graphs Linear system: system comprised entirely of linear equations Consistent system: a linear system in 2 variables with at least one solution Inconsistent system: a linear system with no solution i.e. parallel lines independent: exactly one solution intersecting lines dependent: infinitely many solutions same line

  3. We can solve systems of linear equations by substitution or addition / linear combination. Ex 1) Solve by substitution • 3x – 2(3x – 9) = 0 • 3x – 6x + 18 = 0 • –3x = –18 • x = 6 (6, 9)

  4. Ex 2) Solve by addition • 2(–5) + 3y + 1 = 0 • –9 + 3y = 0 • 3y = 9 • y = 3 + • 11x = –55 • x = –5 (–5, 3)

  5. Ex 3) Solve using substitution and sketch the graph • 13 = 9(2) – y • –5 = –y • 5 = y • (2, 5) • 13 = 9(–1) – y • 22 = –y • –22 = y • (–1, –22) graphing calculator y1 = –3x2 + 12x – 7 y2 = 9x – 13 Find intersections  they match!

  6. Systems are also used to model real world problems Ex 4) An animal trainer has 600 ft of fencing and wants to build an L-shaped pen with a total area of 6800 ft2. The pen is composed of two juxtaposed rectangles of equal dimensions as shown in the diagram. Find the dimensions of the rectangles. x Areas: xy and xy 2xy = 6800 y y A Perimeter: y + x + x + (y – x) + y + x + y = 600 B x A B 4y + 2x = 600 • Solve • 2xy = 6800 • 4y + 2x = 600 •  xy = 3400 •  2y + x = 300 •  x = 300 – 2y  (300 – 2y)y = 3400

  7. Ex 4) cont… (300 – 2y)y = 3400 300y – 2y2 = 3400 2y2 – 300y + 3400 = 0 y2 – 150y + 1700 = 0 Quadratic formula! x(137.65) = 3400 x = 24.70 x(12.35) = 3400 x = 275.30 (y needs to be bigger than x) 24.70 ft × 137.65 ft or 12.35 y = 137.65

  8. Homework #801 Pg 390 #1, 5, 7, 9, 11, 12, 14, 16, 18, 21, 22, 28, 34, 35, 38, 40, 42–45 HW Hint: Simple Interest Formula: I = Prt

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