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3.1 Graph each of the following problems 1. 4. 2. 5. 3. 6. 3.1 A system of linear equations is 2 or more equations that intersect at the same point or have the same solution. You can find the solution to a system of equations in several ways. The one you are going to learn today is to find a solution by graphing. The solution is the ordered pair where the 2 lines intersect. In order to solve a system, you need to graph both equations on the same coordinate plane and then state the ordered pair where the lines intersect. Classifying Systems • Consistent – a system that has at least one solution • Inconsistent – a system that has no solutions • Independent – a system that has exactly one solution • Dependent – a system that has infinitely many solutions Lines intersect at one point: consistent and independent Lines coincide; consistent and dependent Lines are parallel; inconsistent Graph each system and then estimate the solution. GUIDED PRACTICE 2. 4x – 5y = -10 2x – 7y = 4 1. 3x + 2y = -4 x + 3y = 1 3x + 2y = -4 2y = -3x - 4 3 y x2 2 x + 3y = 1 3y = -x + 1 1 1 y x 3 3 4x – 5y = -10 2x – 7y = 4 -5y = -4x -10 -7y = -2x + 4 y 4 x2 5 y 2 4 x 7 7 From the graph, the lines appear to intersect at (–2, 1). From the graph, the lines appear to intersect at (–5, –2). Consistent & Independent Consistent & Independent GUIDED PRACTICE 3. 8x – y = 8 3x + 2y = -16 8x – y = 8 3x + 2y = -16 -y = -8x + 8 2y = -3x - 16 y = 8x - 8 3 y x 8 2 From the graph, the lines appear to intersect at (0, –8). Consistent & Independent Solve the system. Then classify the system as consistent and independent,consistent and dependent, or inconsistent. 2x + y = 4 4x – 3y = 8 2x + y = 1 8x – 6y = 16 4x – 3y = 8 – 3y = -4x + 8 y 4 8 x 3 3 8x – 6y = 16 – 6y = -8x + 16 y 4 8 x 3 3 2x + y = 4 2x + y = 1 y = -2x + 4 y = -2x + 1 (the lines have the same slope) (the equations are exactly the same) The system has infinite solutions the system has no solution consistent and dependent. inconsistent. Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent. A. 2x + 5y = 6 4x + 10y = 12 2x + 5y = 6 5y = -2x + 6 2 6 y x 5 5 B. 4x + 10y = 12 Same equation Infinite solutions Consistent and independent 10y = -4x + 12 y 2 6 x 5 5 3x – 2y = 10 3x – 2y = 10 3x – 2y = 2 3x – 2y = 2 2y = – 3x + 10 2y = – 3x + 2 3 y x5 2 3 y x 1 2 Same slope // lines no solution inconsistent (–1, 3) C. – 2x + y = 5 y=–x+2 – 2x + y = 5 y = 2x + 5 consistent y=–x+2 independent C. – 2x + y = 5 y=–x+2 (–1, 3) – 2x + y = 5 y=–x+2 y = 2x + 5 Is (-1,3) the correct solution? – 2x + y = 5 – 2(-1) + (3)= 5 y=–x+2 3 = – (-1) + 2 2+3=5 ☺ HOMEWORK 3.1 P.156 #3-10 and board work 3=1+2 consistent ☺ independent (3, 3) No solution (-1, 1) Infinite solutions Solve each system of equations by graphing. Indicate whether the system is Consistent- Independent, Consistent-Dependent, or Inconsistent 5. 6. 5. yx4 6. y x 2 y 2x 4 y 2x 5 no solution (-1, 3) 7. 7. Consistent, independent Infinite solutions y 3x 2 2 y 6x 4 11. y 2 x 4 9. 10. 2 x 2 y 10 Consistent, No solutions dependent 11. Inconsistent (2, 1) y 2x 3 y=½x 2x y 4 x y2 y = -2x + 4 y =x-2 15. y 2 3x x 1 y 2 3 6 y = -3x + 2 x y6 3x 3 y 18 y= -x + 6 y = -x + 6 12. x 2y 0 13. (1, 2) 10. x y6 y=x+6 y=x+5 8. y 2x inconsistent, y = 3x - 2 9. 8. Consistent, independent 12.Infinite solutions y 12 x 4 Consistent, dependent 2y 8 x 13. Infinite solutions y = 1/2x + 4 14. Consistent, (2, 0) independent14. Infinite solutions 15. Consistent, independent 16. Consistent, dependent x y2 2 x 2 y 12 y = -x + 2 y = -x + 6 2x 4 y 3 1 x y 1 2 4 y = -2x + 4 y = 6x - 4 Consistent, dependent No solutions 16. inconsistent (1, 1) Consistent, independent