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Graph each of the following systems 22. 3x – 5y=-9 5x -6y= -8 23. 4x + 5y= -2 4y – x = 11 Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent. 36. y= -3x + 19 Y= 2x – 1 Plug the first into the second: -3x+19 = 2x-1 Add 3x: 19 = 5x-1 Add 1: 20 = 5x X=4 Get y: Y = -3*4 + 19 = -12 + 19 = 7 So: X = 4, y = 7 (independent, consistent) 38. y= -4x – 7 Y= 3x Plug the second equation into the first: 3x = -4x – 7 Add 4x: 7x = -7 X = -1 Get y: Y = 3x = 3*-1 = -3 So: X = -1, y = -3 (independent, consistent) Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent. 42. y= x + 4 3y – 5x= 6 Plug the first into the second: 3(x+4)-5x=6 3x+12 – 5x = 6 -2x + 12 = 6 Subtract 12: -2x = -6 X=3 Get y: Y = x+4 = 3+4 = 7 So: X = 3, y = 7, independent, consistent 56. 2x – y= 4 2x – y= 3 Substitute the 2x-y from the first into the second: 4=3 That’s not true, so there are no solutions. inconsistent. 70. x + 3y= 2 -x + y= 1 Subtract 3y from the first: X = 2-3y Plug that into the second: -(2-3y)+y = 1 -2+3y+y = 1 -2+4y = 1 4y = 3 Y = 3/4 Get x: X = 2-3y X = 2-3*3/4 = 2-9/4 = -1/4 So: X = -1/2, y = ¾, independent, consistent Exercises 7.2 Solve each system by addition. 8. x + y= 7 X – y = 9 Add the equations: 2x=16 X=8 Get y: x+y=7 8+y=7 Y = -1 So: X = 8, y = -1, consistent, independent 12. x – 2y= -1 -x + 5y= 4 Add them: 3y = 3 Y=1 Get x: x-2*1 = -1 x-2 = -1 x=1 So: X = 1, y = 1, independent, consistent 16. 3x + 5y= -11 X – 2y= 11 Multiply the second by -3: -3x+6y=-33 Add that to the first: 11y = -44 Y = -4 Get x: 3x + 5*-4 = -11 3x – 20 = -11 3x = 9 X=3 So: X = 3, y = -4, independent, consistent 22. 2x= 2- y 3x + y= -1 Add them: 2x-1 = 3x+2 -1 = x + 2 X = -3 Get y: 3x + y = -1 3*-3 + y = -1 -9 + y = -1 Y=8 So: X = -3, y = 8, consistent, independent Solve each system by the addition method. Determine whether the equations are independent, dependent, or inconsistent. 26. x – y= 3 -6x + 6y = 17 Multiply the first by 6: 6x-6y = -18 Add to the other one: 0 = -1 That’s not true, so no solution, inconsistent. 36. 3/7x + 5/9y= 27 1/9x + 2/7y = 7 Multiply both equations by 63 to clear the fractions: 27x + 35y = 1701 7x + 18y = 441 Now, multiply the first by 7 and the second by -27: 189x + 245y = 11907 -189x - 486y = -11907 Add them: -241y = 0 So y = 0. Get x: 189x = 11907 Divide by 189: x = 63 so: x = 63, y = 0 Independent, consistent 44. 3x – 2.5y= 7.125 2.5x – 3y= 7.3125 Multiply by 16. 48x – 40y = 114 40x – 48y = 117 Multiply the first by 10 and the second by -12: 480x – 400y = 1140 -480x + 576y = -1404 Add: 176y = -264 Y = 1.5 Get x: 3x – 2.5*1.5 = 7.125 X = 1.125 So: x = 1.125, y = -1.5 consistent, independent