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Topic: EQUATIONS

Topic: EQUATIONS. Simple Equations Fractional Equations. Guidelines. Equations must be balanced. You must respect the laws of equations. The goals is to bring variable on the left and number to the right. Coefficient (or number) of variable +1. For example, 1x = 5; just put x = 5.

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Topic: EQUATIONS

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  1. Topic: EQUATIONS Simple Equations Fractional Equations

  2. Guidelines • Equations must be balanced. You must respect the laws of equations. • The goals is to bring variable on the left and number to the right. • Coefficient (or number) of variable +1. For example, 1x = 5; just put x = 5

  3. Examples • #1) Solve: x = 5 + 2 x = 7 • x = 5 (2 + 7) – ( 7 – 3) x = 10 + 35 – 7 + 3 • Remember a minus before a bracket changes the sign of everything in the bracket x = 41

  4. Guidelines con’t • Whatever you do to one side, you must do to another. If you add 5 to one side, you must add 5 to the other side. • If you have a number near the variable, always divide it by that number. For example 2x = 10; divide both by 2. x = 5 • If -5x = 10; divide by -5; x = -2 • When it changes signs, it changes signs

  5. Example • 4x + 1 = 13 4x = 13 - 1 (the 1 changed sides so it changes signs) 4x = 12 (divide both sides by 4) x = 3

  6. Verification • If you want to guarantee that you have the right answer you should verify. To verify: replace the number into the letter in the question. • 4x + 1 = 13 (Original question) & x=3 4 (3) + 1 = 13 12+1=13 13=13 This is true; you have the right answer.

  7. Guideline & Example • 3x – 5 = 10x + 10 • -3x - 10x = 10 + 5 • -13x = 15 • x = -1.15

  8. Another Example – Long Version • 3x – 5 = 8x + 15 • 3x – 8x – 5 + 5 = 8x – 8x + 15 + 5 • -5x = 20 • x = -4

  9. Same example – Short Version • 3x – 5 = 8x + 15 • -5x = 20 • x = -4 • We will continue with the short version ;)

  10. More examples • 4x + 7 = 2x – 11 • 2x = -18 • x = - 9 • Verify! • 4 (-9) + 7 = 2 (-9) – 11 • -36 + 7 = -18 – 11 • - 29 = -29 (You have the right answer)

  11. Another Example • 9x – 5 = 2x + 4 7x = 9 x = 1.29 You can still verify this! 9 (1.29) – 5 = 2 (1.29) + 4 6.61 = 6.58 (rounding error… close enough!)

  12. More Examples • 3x + 5 = 6x + 25 • x = - 6.67 • 4x + 2 = 8x – 31 • X = 8.25 --- are you verifying?)

  13. Reminders • Distributive property • 5(3x + 2) means you multiply 5 by everything in the bracket • 15x + 10 • A minus sign before the bracket changes the sign of everything in the bracket

  14. Now to add some fun – and have them longer • 3 (5x-7) – (2x+8) = 2 (3x-1) • 15x – 21 – 2x – 8 = 6x – 2 • CLEAN IT UP BEFORE MOVING NUMBERS OR LETTERS • 13x – 29 = 6x – 2 • 7x = 27 • x = 3.86 • Verify!

  15. Verify • 3 (5x-7) – (2x+8) = 2 (3x-1) • 3 (5(3.86) – 7) – (2(3.86) - 8 = 2 (3(3.86) – 1) • 3 (12.3) – 7.72 – 8 = 2 (10.58) • 21.18 = 21.16 (rounding error – close enough!)

  16. Quiz # 3 Equations • 1) 5x – 7 = 3x + 7 • 2) 2x – 5 = 4x + 7 • 3) 9x – 2 = - 40 + 5x • 4) 3x – 7 = 8x + 20 • 5) 6x – 1 = 8x + 20

  17. 6) 2 (3x – 7) = 5 (3x – 1) 7) 7(4x + 1) – 5 (3x + 5) = 8x – (3x + 2) 8) 2x – (3x + 1) 9) 2x + 5 10) – 5 – 5 Quiz #3 Con’t

  18. Quiz #3 Equation Solutions • 1) 7 (1a) • 2) – 6 (1b) • 3) -9.5 (1i) • 4) – 5.4 (1o) • 5) -10.5 • 6) -1 (1u) • 7) 2 (1w) • 8) –x – 1 (minus before a bracket!) • 9) 2x + 5 (don’t mix apples & oranges!) • 10) – 10

  19. Fractional Equations • Once again we want to get rid of the fraction • Find the LCD (Lowest Common Denominator) • Multiply every term to get the LCD.

  20. Example • x – 2 = 11 5 3 15 LCD of 5, 3 and 15 is LCD is 15 3x – 10 = 11 3x = 21 X = 7 and then yes… VERIFY

  21. Verify • x – 2 = 11 5 3 15 7/5 – 2/3 = 11 / 15 0.73 = 0.73 it works!

  22. Another example • 2x – 11 = 3x – 5 7 14 28 7 8x – 22 = 3x – 20 5x = 2 x = 0.4

  23. Another Example • 3 – 11x = 5 + 5x 8 12 24 6 9 – 22x = 5 + 20x -42x = -4 X = 0.1

  24. Final Example (Hard) • 3x – 2 – 11x + 8x = 3 – 7x + 1 5 3 30 15 2 15 30 18x – 20 – 11x + 16x = 45 – 14x +1 37x = 66 X = 1.78

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