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Transcript
Solving Equations Using Addition, Subtraction, Multiplication and Division Objectives: • Apply these skills to solve practical problems. • Justify steps used in solving equations. • Use a graphing calculator to check your solutions. To Solve an Equation means... • To isolate the variable having a coefficient of 1 on one side of the equation. • Ex: x = 5 is solved for x. • y = 2x - 1 is solved for y. Addition Property of Equality For any numbers a, b, and c, if a = b, then a + c = b + c. What it means: You can add any number to BOTH sides of an equation and the equation will still hold true. An easy example: • Would you ever We all know that 7 = 7. leave the house with only one shoe on? Does 7 + 4 = 7? NO! • Would you ever put blush on just one But 7 + 4 = 7 + 4. cheek? The equation is still • Would you ever true if we add 4 shave just one side of your face? to both sides. Let’s try another example! 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. What if we see y + (-4) = 9? Recall that y + (-4) = 9 • Check your solution! is the same as y - 4 = 9. • Does 13 4 = 9? Now we can use the • YES! 9=9 and addition property. our solution is y-4=9 correct. +4 +4 y = 13 How about -16 + z = 7? • Remember to always • Check you solution! use the sign in front of the number. • Does -16 + 23 = 7? • Because 16 is negative, we need to add 16 to both sides. • YES! 7 = 7 and our solution is correct. • -16 + z = 7 +16 +16 z = 23 A trick question... -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. 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. 3 Examples: 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! Try these on your own... x + 4 = -10 x – 14 = -5 y – (-9) = 4 3 – y= 7 12 + z = 15 -5 + z = -7 The answers... x = -14 x=9 y = -5 y = -4 z=3 z = -2 Multiplication Property of Equality For any numbers a, b, and c, if a = b, then ac = bc. What it means: You can multiply BOTH sides of an equation by any number and the equation will still hold true. An easy example: We all know that 3 = 3. Would you ever put deodorant under just one arm? Does 3(4) = 3? NO! But 3(4) = 3(4). The equation is still true if we multiply both sides by 4. Would you ever put nail polish on just one hand? Would you ever wear just one sock? Let’s try another example! 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. What do we do with negative fractions? Recall that x x x 5 5 5 x 3. Solve 5 Multiply both sides by -5. • The two negatives will cancel each other out. • The two fives will cancel xeach other out. 3(-5) (-5) 5 • x = -15 • Does -(-15)/5 = 3? 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 zeroand the equation will still hold true. Why did we add c ≠ 0? 2 Examples: 1) 4x = 24 Divide both sides by 4. 4x = 24 4 4 x=6 2) -6x = 18 Divide both sides by -6. -6y = 18 -6 -6 y = -3 • Does 4(6) = 24? YES! • Does -6(-3) = 18? YES! A fraction times a variable: 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 Try these on your own... x=3 7 4w = 16 y=8 -2 2x = 12 3 -2z = -12 3x = 9 -4 The answers... x = 21 w= 4 y = -16 x = 18 z=6 x = -12