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Solving Equations with Variables on Both Sides. 2-4. Contradictions & Identities. Holt Algebra 1. Vocabulary. identity contradiction. An identity is an equation that is true for all values of the variable. An equation that is an identity has infinitely many solutions.

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2-4

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  1. Solving Equations with Variables on Both Sides 2-4 Contradictions & Identities Holt Algebra 1

  2. Vocabulary identity contradiction

  3. An identity is an equation that is true for all values of the variable. An equation that is an identity has infinitely many solutions. A contradiction is an equation that is not true for any value of the variable. It has no solutions.

  4. Identities and Contradictions

  5. Identities and Contradictions WORDS Contradiction When solving an equation, if you get a false equation, the original equation is a contradiction, and it has no solutions. NUMBERS 1 = 1 + 2 1 = 3  ALGEBRA x = x + 3 –x–x 0 = 3 

  6. + 5x + 5x Example 3A: Infinitely Many Solutions or No Solutions Solve 10 – 5x + 1 = 7x + 11 – 12x. 10 – 5x + 1 = 7x + 11 – 12x 10– 5x+ 1 = 7x+ 11– 12x Identify like terms. 11 – 5x = 11 – 5x Combine like terms on the left and the right. Add 5x to both sides. 11 = 11 True statement.  The equation 10 – 5x + 1 = 7x + 11 – 12xis an identity. All values of x will make the equation true. All real numbers are solutions.

  7. –13x–13x Example 3B: Infinitely Many Solutions or No Solutions Solve 12x – 3 + x = 5x – 4 + 8x. 12x – 3 + x = 5x – 4 + 8x 12x– 3+ x = 5x– 4+ 8x Identify like terms. 13x – 3 = 13x – 4 Combine like terms on the left and the right. Subtract 13x from both sides.  –3 = –4 False statement. The equation 12x – 3 + x = 5x – 4 + 8xis a contradiction. There is no value of x that will make the equation true. There are no solutions.

  8. Lesson Quiz Solve each equation. 1. 7x + 2 = 5x + 82. 4(2x – 5) = 5x + 4 3. 6 – 7(a + 1) = –3(2 – a) 4. 4(3x + 1) – 7x = 6 + 5x – 2 5. 6. A painting company charges $250 base plus $16 per hour. Another painting company charges $210 base plus $18 per hour. How long is a job for which the two companies costs are the same? 3 8 all real numbers 1 20 hours

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