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Solving Equations Vocabulary An algebraic equation is an equation that includes one or more variables. An equivalent equations is an equation that have the same solution(s). Isolate – to solve an equation you must isolate the variable (i.e. get the variable alone to one side of the equation). An inverse operation undoes another operation by performing the opposite operation (i.e subtraction is the inverse operation of addition) 2 Properties of Equality – producing equivalent equations Addition Property of Equality: Adding the same number to each side of an equation produces an equivalent equation. x–3=2 x–3+3=2+3 Subtraction Property of Equality: Subtracting the same number to each side of an equation produces an equivalent equation. x+3=2 x+3-3=2-3 3 Properties of Equality – producing equivalent equations Multiplication Property of Equality: Adding the same number to each side of an equation produces an equivalent equation. 𝑥 3 𝑥 3 =2 ∙3=2∙3 Division Property of Equality: Dividing the same number to each side of an equation produces an equivalent equation. 5x = 20 5𝑥 𝟓 = 20 𝟓 4 Solving One-Step Equations 5 Solving an One-Step Equation Solving an equation with subtraction Solving an equation with addition x + 13 = 27 -7 = b - 3 isolate the variable x + 13 – 13 = 27 – 13 undo addition by subtracting the same number from each side -7 + 3 = b – 3 + 3 undo subtraction by adding the same number from each side x = 14 -4 = b Simplify each side of the equation isolate the variable Simplify each side of the equation Check it! Check it! x + 13 = 27 14 + 13 = 27 27 = 27 Substitute the answer into original equation to check it. -7 = b - 3 -7 = -4 - 3 -7 = -7 Substitute the answer into original equation to check it. 6 Solving an One-Step Equation Solving an equation with multiplication Solving an equation with division 4x = 28 isolate the variable 4𝑥 4 undo multiplication by dividing the same number from each side 𝒙 𝟒 𝒙 𝟒 Simplify each side of the equation x = -36 = 28 4 x=7 Check it! 4x = 28 4 ∙ 7 = 28 28 = 28 Substitute the answer into original equation to check it. = -9 isolate the variable ∙ 𝟒 = -9 ∙ 4 undo division by multiplying the same number from each side Check it! 𝒙 = -9 𝟒 −𝟑𝟔 = -9 𝟒 -9 = -9 Simplify each side of the equation Substitute the answer into original equation to check it. 7 Solving an One-Step Equation Solving an equation using reciprocal 𝟒 𝒎 𝟓 isolate the variable = 28 𝟓 𝟒 5 𝒎 = (28) 𝟒 𝟓 4 m = 35 Multiply each side by 5/4, the reciprocal of 4/5 Simplify each side of the equation Check it! 8 Solving Two and Multi -Step Equations 9 Understanding Two-Step To solve two-step equations, you use the properties of equality and inverse operations to form a series of simpler equivalent equations. You can use the properties of equality repeatedly to isolate the variable. Multi-Step To solve two-step equations, you use the properties of equality, inverse operations, and properties of real numbers to form a series of simpler equivalent equations. You use the properties until you isolate the variable. 10 You can undo the operations in the reverse order of the order of operations. 2x + 3 = 15 2x + 3 – 3 = 15 – 3 subtract 3 from each side 2x = 12 simplify 2𝑥 2 Divide each side by 2 = 12 2 x=6 Simplify Check 2x + 3 = 15 2(6) + 3 = 15 15 = 15 11 When one side of an equation is a fraction with more than one term in the numerator, you can still undo division by multiplying each side by the denominator. 𝑥 −7 3 3 = −12 𝑥 −7 3 = 3 (−12) Multiply each side by 3 x – 7 = -36 Simplify x – 7 + 7 = -36 + 7 Add 7 to each side x = -29 Simplify 12 Using Deductive Reasoning When you use deductive reasoning, you must state your steps and your reason for each step using properties, definitions, or rules. For example: Steps Reasons -t + 8 = 3 Original equation -t + 8 - 8 = 3 – 8 Subtraction Property of Equality -t = -5 Use subtraction to simplify -1t = -5 −𝟏𝒕 −𝟓 = −𝟏 −𝟏 t=5 Multiplicative Property of -1 Division Property of Equality Use division to simplify 13 Solving a Equation by Combining Like Terms 5 = 5m – 23 + 2m Original Equation 5 = 5m + 2m – 23 Commutative Property of Addition 5 = 7m – 23 Combine like terms 5 + 23 = 7m – 23 + 23 Add 23 to each side 28 = 7m 𝟐𝟖 𝟕𝒎 = 𝟕 𝟕 4=m Simplify Divide each side by 7 Simplify 14 Solving an Equation Using the Distributive Property Make the equation easier by removing the grouping symbols first! -8(2x – 1) = 36 Original Equation -16x + 8 = 36 Distributive Property -16 +8 -8 = 36 – 8 Subtract 8 from both sides -16x = 28 −𝟏𝟔𝒙 𝟐𝟖 = −𝟏𝟔 −𝟏𝟔 𝟕 x=− 𝟒 Simplify Divide each side by -16 Simplify 15 Solving an Equation That Contains Fractions You can use different methods to solve equations that contain fractions. Method 1 Method 2 Write the like terms by using common denominator and solve. Clear the fraction from the equation 𝟑𝒙 𝒙 − = 𝟏𝟎 𝟒 𝟑 𝟑 𝟏 𝒙 − 𝒙 = 𝟏𝟎 𝟒 𝟑 𝟗 𝟒 𝒙− 𝒙 = 𝟏𝟎 𝟏𝟐 𝟏𝟐 Original equation 𝟓 𝒙 = 𝟏𝟎 𝟏𝟐 𝟏𝟐 𝟓 𝟏𝟐 𝒙 = 𝟏𝟎 𝟓 𝟏𝟐 𝟓 Combine like terms x = 24 Simplify Rewrite the fraction Write the fraction using the common denominator 12. Multiply each side by the reciprocal 𝟓 of𝟏𝟐 𝟏𝟐 , 𝟓 𝟑𝒙 𝒙 − = 𝟏𝟎 𝟒 𝟑 𝟑𝒙 𝒙 𝟏𝟐 − = 𝟏𝟐(𝟏𝟎) 𝟒 𝟑 Original equation 𝟑𝒙 𝒙 𝟏𝟐 − 𝟏𝟐 = 𝟏𝟐(𝟏𝟎) 𝟒 𝟑 Distributive property 9x – 4x = 120 Multiply 5x = 120 Combine like terms x = 24 Divide each side by 16 5 and simplify. Multiply each side by a common denominator, 12 Solving an Equation That Contains Decimals You can clear decimals from an equation by multiplying by a power of 10. First, find the greatest number of digits to the right of any decimal point, and then multiply by 10 raised to that power. 3.5 – 0.02x = 1.24 Original equation 100(3.5 – 0.02x) = 100(1.24) Multiply each side by 103, or 100 350 – 2x = 124 Distributive Property 350 – 2x – 350 = 124 – 350 Subtract 350 from each side -2x = -226 −𝟐𝒙 −𝟐𝟐𝟔 = −𝟐 −𝟐 x = 113 Simplify Divide each side by -2 Simplify 17 Solving Equations with Variables on Both Sides 18 How To Get Started There are variables terms on both sides of the equation. Decide which variable term to add or subtract to get the variable on one side only. To solve equations with variables on both sides, you can use the properties of equality and inverse operations to write a series of simpler equivalent equations. 19 Solving an Equation with Variables on Both Sides 5x + 2 = 2x + 14 Original equation 5x + 2 – 2x = 2x +14 – 2x Subtract 2x from each side 3x + 2 = 14 Simplify 3x + 2 – 2 = 14 Subtract 2 from each side 3x = 12 Simplify 𝟑𝒙 𝟏𝟐 = 𝟑 𝟑 Divide each side by 3 x=4 Simplify 20 Solving an Equation With Grouping Symbols 2(5x – 1) = 3(x + 11) Original equation 10x – 2 = 3x + 33 Distributive Property 10x – 2 – 3x = 3x + 33 – 3x Subtract 3x from each side 7x – 2 = 33 Simplify 7x – 2 + 2 = 33 + 2 Add 2 to each side 7x = 35 𝟕𝒙 𝟑𝟓 = 𝟕 𝟕 x=5 Simplify Divide each side by 7 Simplify 21 Identities and Equations with No Solutions An equation that is true for every possible value of An equation has no solution if there is no value of the the variable is an identity. variable that make the equation true. The equation For example: x + 1 = x + 1 is an identity. x +1 = x +2 has no solution. 10x + 12 = 2(5x + 6) Original equation 9m – 4 = -3m + 5 + 12m Original equation 10x + 12 = 10x + 12 The Distributive Property 9m – 4 = 9m + 5 Combine like terms 9m – 4 – 9m = 9m + 5 – 9m Subtract 9m from each side -4 = 5 Simplify Because 10x + 12 = 10x + 12 is always true, there are infinitely many solutions of the equation (x can equal any thing and it still will remain true). The original equation is an identity. Because -4 ≠ 5, the original equation has no solution 22 When you solve an equation, you use reasoning to select properties of equality that produce simpler equivalent equations until you find a solution. The steps below provide a general guideline for solving equations. Concept Summary Solving Equations Step 1 Use the Distributive Property to remove any grouping symbols. Use properties of equality to clear decimals and fractions. Step 2 Combine like terms on each side of the equations. Step 3 Use the properties of equality to get the variables terms on one side of the equation and the constants on the other. Step 4 Use the properties of equality to solve for the variable. Step 5 Check your solution in the original equation 23 Do You Know How? Solve each equation. Check your answer. Solving one-step equations Solving two-step equations 1. x + 7 = 3 5. 5x + 12 = -13 2. 9 = m – 4 6. 6 = 3. 5y = 24 4. You have already read 117 pages of a book. You are one third of the way through the book. Write and solve an equation to find the number of pages in the book. 7. 𝑦−1 4 𝑚 7 −3 = −2 8. -x – 4 = 9 9. The junior class is selling ganola bars to raise money. They purchase 1250 granola bars and paid a delivery fee of $25. The total cost, including the delivery fee, was $800. What was the cost of 24 each granola bar? Do You Know How? Solve each equation. Check your answer. Solving mult-step equations Solving equations with variables on both sides 10. 7p + 8p – 12 = 59 15. 3x + 4 = 5x – 10 11. -2(3x + 9) = 24 16. 5(y – 4) = 7(2y + 1) 12. 2𝑚 7 + 3𝑚 14 =1 17. 2a + 3 = 1 2 6 + 4𝑎 13. 1.2 = 2.4 – 0.6x 18. 4x – 5 = 2(2x + 1) 14. There is a 12-ft fence on one side of a rectangular garden. The gardener has 44 ft of fencing to enclose the other three sides. What is the length of the garden’s longer dimension? (draw a 19. Pristine Printing will print business cards for $0.10 each plus a setup charge of $15. The Printing Place offers business cards for $0.15 each with a setup charge of $10. What number of business cards costs the same from either printer? 25 diagram, if it helps) Do You UNDERSTAND? Solving one-step equations Which property of equality would you use to solve each equation? Why? See pg. 3-4 20. 3 + x = -34 21. 2x = 5 22. x – 4 = 9 Solving two-step equations What properties of equality would you use to solve each equation? What operation you perform first? Explain 𝑠 4 25. -8 = + 3 26. 2x – 9 = 7 𝑥 3 23. 𝑥/7=9 27. 24. Write a one-step equation. Then write two equations that are equivalent to your equation. How can you prove that all three equations are equivalent? 28. -4x + 3 = -5 −8=4 𝑑−3 29. Can you solve the equation = 6 by 5 adding 3 before multiplying by 5? 26 Explain Do You UNDERSTAND? Solving multi-step equations Identities and Equations with no Solutions Explain how you would solve each equation. Match each equation with the appropriate number solutions. 30. 1.3 + 0.5x = -3.41 34. 3y – 5 = y + 2y – 9 A. infinitely many 31. 7(3x – 4) = 49 35. 2y + 4 = 2(y + 2) B. one solution 2 7 32. − 9 𝑥 − 4 = 18 36. 2y – 4 = 3y – 5 C. no solution 33. Ben solves the equation -24 = 5(g + 3) by 37. A student solved an equation and found first dividing each side by 5. Amelia that the variable was eliminated in the solves the equation by using the process of solving the equation. How would Distributive Property. Whose method do the student know whether the equation is you prefer? Explain an identity or an equation with no solution? 27