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Equations

Equations An equation is a statement that two expressions are equal. Algebraic equations contain one or more variables. First-degree equations in one variable.

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Equations

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  1. Equations An equation is a statement that two expressions are equal. Algebraic equations contain one or more variables. First-degree equations in one variable A first-degree equations is an equation that contains only one variable and the variable has an exponent of 1 (Also called a linear equation). To solve first-degree equations in one variable, we will need the following properties. (for all real numbers a, b, and c, c  0) Procedure for Solving Equations of the form ax + b = c Step 1. Isolate the term with the variable by adding the opposite of the constant term, b,to both sides (Addition Property of Equality). Step 2. Multiply both sides by the reciprocal of ‘a’ on both sides (Multiplication property of equality). Step 3. Simplify both sides of the equation. Step 4. Write and Check the solution. Note: There are two formats for solving equations, horizontal and vertical. The text shows only the horizontal format. This lesson will show both formats. Choose the method you are most comfortable with.

  2. Example 1. Solve and Check: 4x + 2 = 22. (Horizontal Format) Solution: 5 5 Example 1. Solve and Check: 4x + 2 = 22. (Vertical Format)  Solution: Check: Replace x with 5. Your Turn Problem #1 Solve and Check: 7x + 4 = 31 Isolate variable term by adding -2 to both sides. Simplify each side.

  3. Choose the method you are comfortable with. The examples will usually use the vertical format. Solution: 4 Answer: (Symmetric Property: if a=b , then b=a) Check:  Your Turn Problem #2 Add 2 to both sides. Simplify each side.

  4. Procedure: Solving Equations of the form ax + b = cx + d Step 1. Add the opposite of the variable term on the right to both sides, then simplify. Step 2. Add the opposite of the constant term on the left to both sides, then simplify. Step 3. Multiply both sides by the reciprocal of the variable term (it should be on the left). Step 4. Write and Check the solution. (horizontal format) (vertical format) Solution: 6 6 Answer: Answer: Check: Your Turn Problem #3 Solve and Check:  Answer:

  5. Equations may not always be written in the forms given on the previous slides. Therefore, always simplify both sides of the equation before performing the necessary steps. Use the distributive property to get rid of parenthesis. Solution: Answer: Check: Your Turn Problem #4 Solve and Check: Answer:  Then simplify left and right hand sides Proceed as in previous examples.

  6. Use the distributive property to get rid of parentheses. Solution: Answer: Check:  Your Turn Problem #5 Simplify left and right hand side. Proceed as in previous examples. In this case, get x on the right hand side.

  7. Procedure: Translating Sentences into Equations and Solving Step 1. Assign a variable to the unknown quantity (What is the problem asking you to find?) Step 2. Translate the sentence into a mathematical expression. Step 3. Solve the equation using the steps from the preceding slides. Example 6. Translate and solve: If 8 is subtracted from 3 times a certain number, the result is 34. Find the number. Solution: Let x represent the unknown number. Translated: Translation: 8 subtracted from: 3 times a number: The result is 34: The number is 14. Your Turn Problem #6 Then solve:

  8. Integer Problems The next type of application problems to consider are “consecutive integer” problems. Consecutive integers are integers that follow in a sequence, each number is 1 more than the previous number. Example of consecutive integers: 17, 18, and 19. Consecutive even integers are integers that follow in a sequence where each number is 2 more that the previous number. Example of consecutive even integers: 12, 14, and 16. Consecutive odd integers are integers that also follow in a sequence where each number is 2 more that the previous number. Example of consecutive odd integers: -7, -5, and -3. Consecutive Integer Problems Procedure: To find the consecutive integers: 1. Let x be the first integer. 2. Let x+1 be the second integer. 3. Let x+2 be the third integer. Continue this process if asked to find more than three integers. 4. Translate into a mathematical equation and solve. 5. Check the answer. Note some facts from the examples above. Consecutive integers increase by one. Consecutive odd and even integers increase by two. The key to solving application problems is to follow the given “setups” for that kind of problem. Please follow all steps and procedures as shown. This will help you to obtain the equation and solution.

  9. Solve: Setup: First integer: x Second integer: x+1 Third integer: x+2 Translated: Check: Answer: The integers are –23, -22, and -21 Your Turn Problem #7 Find three consecutive integers whose sum is –36. Example 7: Find three consecutive integers whose sum is -66. Once x is found, the next step is to write the answer.

  10. Consecutive Even Integers Procedure: To find consecutive even integers: 1. Let x be the first even integer. 2. Let x+2 be the second even integer. 3. Let x+4 be the third even integer. etc. 4. Translate into a mathematical equation and solve. 5. Check the answer. Example 8. Find two consecutive even integers such that three times the first equals twice the second. Translated: Setup: First integer: x Second integer: x+2 Solve: Answer: The integers are 4 and 6. Check: Three times the first equals twice the second. Your Turn Problem #8 Once x is found, the next step is to write the answer. Answer: 2, 4, and 6

  11. Consecutive Odd Integers Procedure: To find consecutive odd integers: 1. Let x be the first odd integer. 2. Let x+2 be the second odd integer. 3. Let x+4 be the third odd integer. etc. 4. Translate into a mathematical equation and solve. 5. Check the answer. Example 9. Seven times the first of two consecutive odd integers is five times the second. Find the two odd integers. Solve: Translated: First integer: x Second integer: x+2 Setup: Check: Seven times the first equals twice the second. Answer: The integers are 5 and 7 Your Turn Problem #9 Find three consecutive odd integers such that three times the middle integer is one more than the sum of the first and third. Once x is found, the next step is to write the answer. The end. B.R. 12-07-06 Answer: -1, 1, 3

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