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Solving Linear Equations

Solving Linear Equations

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Solving Linear Equations

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  1. Solving Linear Equations

  2. To Solve an Equation means... • To isolate the variable having a coefficient of 1 on one side of the equation. Examples • x = 5 is solved for x. • y = 2x - 1 is solved for y.

  3. Solving Equations Using Addition and Subtraction

  4. Addition Property of Equality What it means: For any numbers a, b, and c, if a = b, then a + c = b + c. You can add any number to BOTH sides of an equation and the equation will still hold true.

  5. We all know that 7 =7. Does 7 + 4 = 7? NO! But 7 + 4 = 7 + 4. The equation is still true if we add 4 to both sides. An easy example:

  6. x - 6 = 10 Add 6 to each side. x - 6 = 10 +6 +6 x = 16 Always check your solution!! The original problem is x - 6 = 10. Using the solution x=16, Does 16 - 6 = 10? YES! 10 = 10 and our solution is correct. Let’s try another example!

  7. Recall that y + (-4) = 9 is the same as y - 4 = 9. Now we can use the addition property. y - 4 = 9 +4 +4 y = 13 Check your solution! Does 13 - 4 = 9? YES! 9=9 and our solution is correct. What if we see y + (-4) = 9?

  8. Remember to always use the sign in front of the number. Because 16 is negative, we need to add 16 to both sides. -16 + z = 7 +16 +16 z = 23 Check you solution! Does -16 + 23 = 7? YES! 7 = 7 and our solution is correct. How about -16 + z = 7?

  9. -n - 10 = 5 +10 +10 -n = 15 Do we want -n? NO, we want positive n. If the opposite of n is positive 15, then n must be negative 15. Solution: n = -15 Check your solution! Does -(-15)-10=5? Remember, two negatives = a positive 15 - 10 = 5 so our solution is correct. A trick question...

  10. Subtraction Property of Equality • For any numbers a, b, and c, if a = b, then a - c = b - c. What it means: • You can subtract any number from BOTH sides of an equation and the equation will still hold true.

  11. 1) x + 3 = 17 -3 -3 x = 14 Does 14 + 3 = 17? 2) 13 + y = 20 -13 -13 y = 7 Does 13 + 7 = 20? 3) z - (-5) = -13 Change this equation. z + 5 = -13 -5 -5 z = -18 Does -18 -(-5) = -13? -18 + 5 = -13 -13 = -13 YES! 3 Examples:

  12. Try these on your own...

  13. The answers...

  14. Solving Equations Using Multiplication and Division

  15. We all know that 3 = 3. Does 3  4 = 3? NO! But 3  4 = 3  4. The equation is still true if we multiply both sides by 4. An easy example:

  16. x = 4 2 Multiply each side by 2. 2  x= 4  2 2 x = 8 Always check your solution!! The original problem is x = 4 2 Using the solution x = 8, Is x/2 = 4? YES! 4 = 4 and our solution is correct. Let’s try another example!

  17. The two step method: Ex: 2x = 4 3 1. Multiply by 3. (3)2x = 4(3) 3 2x = 12 2. Divide by 2. 2x = 12 2 2 x = 6 The one step method: Ex: 2x = 4 3 1. Multiply by the RECIPROCAL. (3)2x = 4(3) (2) 3 (2) x = 6 A fraction times a variable:

  18. The two negatives will cancel each other out. The two fives will cancel each other out. (-5) (-5) x = -15 Does -(-15)/5 = 3? What do we do with negative fractions? Recall that Solve . Multiply both sides by -5.

  19. Try these on your own...

  20. Division Property of Equality • For any numbers a, b, and c (c ≠ 0), if a = b, then a/c = b/c What it means: • You can divide BOTH sides of an equation by any number - except zero- and the equation will still hold true.

  21. 1) 4x = 24 Divide both sides by 4. 4x = 24 4 4 x = 6 Does 4(6) = 24? YES! 2) -6x = 18 Divide both sides by -6. -6y = 18 -6 -6 y = -3 Does -6(-3) = 18? YES! 2 Examples:

  22. The answers...

  23. Solving Equations with the Variable on Both Sides

  24. To solve these equations, Use the addition or subtraction property to move all variables to one side of the equal sign. Solve the equation using the methods we mentioned.

  25. 1) 6x - 3 = 2x + 13 -2x -2x 4x - 3 = 13 +3 +3 4x = 16 4 4 x = 4 Be sure to check your answer! 6(4) - 3 =? 2(4) + 13 24 - 3 =? 8 + 13 21 = 21 Let’s see a few examples:

  26. 2) 3n + 1 = 7n - 5 -3n -3n 1 = 4n - 5 +5 +5 6 = 4n 4 4 Reduce! 3 = n 2 Check: 3(1.5) + 1 =? 7(1.5) - 5 4.5 + 1 =? 10.5 - 5 5.5 = 5.5 Let’s try another!

  27. 3) 5 + 2(y + 4) = 5(y - 3) + 10 Distribute first. 5 + 2y + 8 = 5y - 15 + 10 Next, combine like terms. 2y + 13 = 5y - 5 Now solve. (Subtract 2y.) 13 = 3y - 5 (Add 5.) 18 = 3y (Divide by 3.) 6 = y Check: 5 + 2(6 + 4) =? 5(6 - 3) + 10 5 + 2(10) =? 5(3) + 10 5 + 20 =? 15 + 10 25 = 25 Here’s a tricky one!

  28. Let’s try one with fractions! • Steps: • Multiply each term • by the least common • denominator (8) to • eliminate fractions. • Solve for x. • Add 2x. • Add 6. • Divide by 6. 4) 3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x so x = 3/2

  29. 6(4 + y) - 3 = 4(y - 3) + 2y 24 + 6y - 3 = 4y - 12 + 2y 21 + 6y = 6y - 12 - 6y - 6y 21 = -12 Never true! 21 ≠ -12 NO SOLUTION! 3(a + 1) - 5 = 3a - 2 3a + 3 - 5 = 3a - 2 3a - 2 = 3a - 2 -3a -3a -2 = -2 Always true! We write IDENTITY. Two special cases:

  30. Try a few on your own: • 9x + 7 = 3x - 5 • 8 - 2(y + 1) = -3y + 1 • 8 - 1 z = 1 z - 7 2 4

  31. x = -2 y = -5 z = 20 The answers: