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Copyright © Cengage Learning. All rights reserved. 6 Equations and Formulas

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Copyright © Cengage Learning. All rights reserved. Equations 6.1

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In technical work, the ability to use equations and formulas is essential. A variable is a symbol (usually a letter of the alphabet) used to represent an unknown number. An algebraic expression is a combination of numbers, variables, symbols for operations (plus, minus, times, divide), and symbols for grouping (parentheses or a fraction bar). Examples of algebraic expressions are 4x – 9, 3x 2 + 6x + 9, 5x(6x + 4),2x +5/ -3x Equations

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An equation is a statement that two quantities are equal. The symbol “=” is read “equals” and separates an equation into two parts: the left member and the right member. For example, in the equation 2x + 3 =11 the left member is 2x + 3 and the right member is 11.

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Other examples of equations are x – 5 = 6, 3x = 12, 4m + 9 = 3m – 2, To solve an equation means to find what number or numbers can replace the variable to make the equation a true statement. In the equation 2x + 3 = 11, the solution is 4. That is, when x is replaced by 4, the resulting equation is a true statement. 2x + 3 = 11 Equations

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Let x = 4: 2(4) + 3 = 11 8 + 3 = 11 A replacement number (or numbers) that produces a true statement in an equation is called a solution or a root of the equation. Note that replacing x in the equation above with any number other than 4, such as 5, results in a false statement. 2x + 3 = 11 Equations ? True

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Let x = 4: 2(4) + 3 = 11 8 + 3 = 11 A replacement number (or numbers) that produces a true statement in an equation is called a solution or a root of the equation. Note that replacing x in the equation above with any number other than 4, such as 5, results in a false statement. 2x + 3 = 11 Equations ? True

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2. Addition & Subtraction Property of Equality: If the same quantity is added or subtracted from both sides of an equation, the resulting equation is equivalent to the original equation. Examples: Solve x + 5 = 2.Solve x – 4 = 7 x + 5 – 5 = 2 – 5 x – 4 + 4 = 7 + 4 x = –3x = 11 Equations Subtract 5 from both sides. Add 4 to both sides

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3. Multiplication Property of Equality: If both sides of an equation are multiplied by the same (nonzero) quantity, the resulting equation is equivalent to the original equation. Example: Solve Equations Multiply both sides by 9.

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4. Division Property of Equality: If both sides of an equation are divided by the same (nonzero) quantity, the resulting equation is equivalent to the original equation. Example: Solve 4x = 20. x = 5 Basically, to solve a simple equation, use one of the rules and use a number that will undo what has been done to the variable. Equations Divide both sides by 4.

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Solve: x + 3 = 8. Since 3 has been added to the variable, use Rule 2 and subtract 3 from both sides of the equation. x + 3 = 8 x + 3 – 3 = 8 – 3 x + 0 = 5 x = 5 Example 1 Subtract 3 was chosen to undo adding 3.

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A check is recommended, since an error could have been made. To check, replace the variable in the original equation by 5, the apparent root, to make sure that the resulting statement is true. Check: x + 3 = 8 5 + 3 = 8 Thus, the root is 5. Example 1 True cont'd

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Some equations have more than one operation indicated on the variable. For example, the equation 2x + 5 = 6 has both addition of 5 and multiplication by 2 indicated on the variable. Use the following procedure to solve equations like this. Note: This is the reverse of order of operations! Equations

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Solve: 2x + 5 = 6. 2x + 5 – 5 = 6 – 5 2x = 1 The apparent root is Example 6 Subtract 5 from both sides. Divide both sides by 2.

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Check: 2x + 5 = 6 2 + 5 = 6 1 + 5 = 6 Thus, the root is Example 6 ? True cont'd

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Copyright © Cengage Learning. All rights reserved. Equations with Variables in Both Members 6.2

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To solve equations with variables in both members (both sides), such as 3x + 4 = 5x – 12 do the following: First, add or subtract either variable term from both sides of the equation. Note: Choosing the smaller of the two variables to remove will usually avoid working with negative numbers! 3x + 4 = 5x – 12 3x + 4 – 3x = 5x – 12 – 3x 4 = 2x – 12 Equations with Variables in Both Members Subtract 3x from both sides. Note Choosing the smaller term to Subtract avoids negative numbers

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Then take the number that appears on the side of the variable and add it to, or subtract it from, both sides. Solve the resulting equation. 4 = 2x – 12 4 + 12 = 2x – 12 + 12 16 = 2x 8 = x Equations with Variables in Both Members Add 12 to both sides. Divide both sides by 2.

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This equation could also have been solved as follows: 3x + 4 = 5x – 12 3x + 4 – 5x = 5x – 12 – 5x –2x + 4 = –12 –2x + 4 – 4 = –12 – 4 –2x = –16 x = 8 Equations with Variables in Both Members Subtract 5x from both sides. Subtract 4 from both sides. Divide both sides by –2.

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Solve: 5x – 4 = 8x – 13. 5x – 4 – 8x = 8x – 13 – 8x –3x – 4 = –13 –3x – 4 + 4 = –13 + 4 –3x = –9 x = 3 Example 1 Subtract 8x from both sides. Add 4 to both sides. Divide both sides by –3.

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Check: 5x – 4 = 8x – 13 5(3) – 4 = 8(3) – 13 15 – 4 = 24 – 13 11 = 11 Therefore, 3 is a root. Example 1 cont’d True ? ?

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Copyright © Cengage Learning. All rights reserved. Equations with Parentheses 6.3

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To solve an equation having parentheses in one or both members, always remove the parentheses first. Then combine like terms. Equations with Parentheses

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Solve: 5 – (2x – 3) = 7. 5 – 2x + 3 = 7 8 – 2x = 7 8 – 2x – 8 = 7 – 8 –2x = –1 = x = Example 1 Remove parentheses. Combine like terms. Subtract 8 from both sides. Note: 8 is a positive number so we do the opposite or subtract. Divide both sides by –2.

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Check: 5 – (2x – 3) = 7 = 7 5 – (1 – 3) = 7 5 – (–2) = 7 Therefore, is the root. Example 1 cont’d ? ? True

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Group Problems for Practice Page 277 Chapter Test 1-11 odds Review Video: http://www.youtube.com/wa tch?v=6ZCWeZr7420 http://www.youtube.com/wa tch?v=6ZCWeZr7420

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