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2.4 Solving Equations with Variables on Both Sides:

No Solution: an equation has no solution if there is no value to make the equation TRUE. 2.4 Solving Equations with Variables on Both Sides:. Infinitely Many Solutions: An equation that is true for any and every possible value. Identity: an equation that has infinitely many solutions.

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2.4 Solving Equations with Variables on Both Sides:

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  1. No Solution: an equation has no solution if there is no value to make the equation TRUE. 2.4 Solving Equations with Variables on Both Sides: Infinitely Many Solutions: An equation that is true for any and every possible value. Identity: an equation that has infinitely many solutions.

  2. GOAL:

  3. We can find the solution to equations that have variables on both sides of the equal sign by using inverse operations and moving the smallest coefficient to the other side of the equal sign: Ex: Solve 2x – 3 = x+5 Isolate the variable with biggest coefficient + 3 +3 2x = x+8 -x -x x = 8

  4. Ex: Solve 2x – 3 = x+5 + 3 +3 2x = x+8 -x -x x = 8 Check: 2( ) -3 = ( ) + 5 2(8)-3 = (8)+5 16 – 3 = 8+5 13=13

  5. REAL-WORLD: A dance studio charges $50 sign-up fee and $65 per day to take all dance classes. Another studio charges a $90 sign-up fee and only $45 per day to take all classes. For what number of days is the cost of the two dance studios the same?

  6. SOLUTION: Using the given info we have: Studio 1  $50 sign-up fee +50 Studio 1  $65 per day  65x Studio 2  $90 sign-up fee  +90 Studio 2  $45 per day  45x Equal  65x + 50 = 45x + 90

  7.  65x + 50 = 45x + 90 65x + 16 = 45x + 90 Like terms on same side of equ. -45x -45x 20x + 16 = 90 Inverse of add -16 -16 20x = 74 Inverse of multiply 20x /20= 74/20 x = 4 days

  8. YOU TRY IT: What is the solution of 5X – 1 = X + 15?

  9. Solving equations with Distributive Property: To solve equations that include distributive property, we must distribute first, then isolate: Ex: What is the solution of 4(2y+1)=2(y -13)?

  10. Solution: 4(2y+1)= 2(y-13) Distributive 4 and 2 4(2y) +4(1)= 2(y) – 2(13) 8y + 4 = 2y-26 Multiplication Inverse of +4 (addition) – 4 –4 8y = 2y - 30 Move the smallest2y -2y -2y 6y = -30 _____ ____ 6 6 Inverse of multiplication y = - 5

  11. Check: 4(2y+1) = 2(y-13) 4(2( )+1) = 2(( )-13) 4(2(-5)+1) = 2((-5)-13) 4(-10+1) = 2(-5-13) 4(-9) = 2(-18) - 36 = - 36

  12. YOU TRY IT: What is the solution of: ?

  13. Solution: 4(2y+1)= 2(y-13) Distributive 4 and 2 4(2y) +4(1)= 2(y) – 2(13) 8y + 4 = 2y-26 Multiplication Inverse of +4 (addition) – 4 –4 8y = 2y - 30 Move the smallest2y -2y -2y 6y = -30 _____ ____ 6 6 Inverse of multiplication y = - 5

  14. Note: Whenever we solve for an equation for a given variable we might get ONE solution, Infinitely many solutions or NO solutions at all.

  15. ONE SOLUTION: What is the solution of 3(5b-2)= 6 +12b?

  16. Solution: 3(5b-5)= -6+12b Distributive 3 3(5b) -3(5)= –6+12b 15b-15= -6+12b Multiplication +15 +15 Inverse of subtraction 15b = 12b + 15 Inverse of multiplication - 12b -12b 3b = 15 3b/3 = 15/3 Thus b = 5 is our one solution.

  17. YOU TRY IT: What is the solution of: 2a + 3 = a + 10?

  18. INFINITELY MANY SOLUTIONS: What is the solution of 3(4b-2)= -6 +12b?

  19. Solution: 3(4b-2)= -6+12b Distributive 3 3(4b) -3(2)= –6+12b 12b-6 = -6+12b Multiplication +6 +6 Inverse of subtraction 12b = 12b Inverse of multiplication - 12b -12b 0 = 0 Since 0 will always be 0, we have infinite solutions.

  20. YOU TRY IT: What is the solution of: 2a + 3 = ½ (6+4a)?

  21. NO Solution: What is the solution of 2x + 7= -(3 - 2x)?

  22. Solution: 2x + 7 = -1(3– 2x) Distributive -1 2x + 7 = –3 + 2x 2x + 7= - 3 + 2x Multiplication Inverse of subtraction + 3 + 3 2x +10 = 2x Move the smallest2x -2x -2x 10 = 0 Since 10 will never equals 0, there is NO solution.

  23. YOU TRY IT: What is the solution of: 3d + 4 =2 + 3d – ½ ?

  24. VIDEOS: Multi-Step Equations Multi-Step https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/v/why-we-do-the-same--thing-to-both-sides-multi-step-equations https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/v/adding-and-subtracting-the-same-thing-from-both-sides

  25. CLASS WORK: Pages: 105 – 106 Problems: As many as it takes you to master the concept.

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