Lesson 5

Steps in Solving Equations

  • Let’s recall steps in solving equations

5.1: Explaining Equivalent Expressions

Explain or show why each of these equations is equivalent to \(7(x-15) + 3 = 8\).

  1. \(7x - 105 + 3 = 8\)
  2. \(7(x-15) - 5 = 0\)
  3. \(7x - 102 - 8 = 0\)

5.2: Checking Work

Here is Clare’s work to solve some equations. For each problem, do you agree or disagree with Clare’s work? Explain your reasoning.

  1. \(2(x-1)+4 = 3x - 2\)
    \(2x - 2 + 4 = 3x - 2\)
    \(2x + 2 = 3x - 2\)
    \(2x = 3x\)
    \(\text{-}x = 0\)
    \(x = 0\)
  2. \(3(x-1) = 5x + 6\)
    \(3x - 1 = 5x + 6\)
    \(\text{-}1 = 2x + 6\)
    \(\text{-}7 = 2x\)
    \(-3.5 = x\)
  3. \((x-2)(x+3) = x+10\)
    \(x^2 + x - 6 =x + 10 \)
    \(x^2 - 6 = 10\)
    \(x^2 = 16\)
    \(x = 4\)

5.3: Row Game: Rewriting Equations

Work independently on your column. Partner A completes the questions in column A only and partner B completes the questions in column B only. Your answers in each row should match. Work on one row at a time and check if your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find any mistakes.

Partner A: Write an equivalent equation so that the given condition is true.

  1. \(5x+10 = -35\)

    • The expression on the right side is 0

  2. \(x^2 - 9x = 42\)

    • The left side is a product

  3. \(x(x+3) + 9 = 1\)

    • The right side is 0

  4. \(8(x+1) = 5x\)

    • The left side is 0 and there are no parentheses

  5. \(11+x = \frac{12}{x}\)

    • The equation is quadratic and the right side is zero.

  6. \((3x-5)(x-2) = 0\)

    • One side of the equation has a term with \(3x^2\)

  7. \(4x^2 - 4 = 8\)

    • The right side is 0 and the left side is a product

Partner B: Write an equivalent equation so that the given condition is true.

  1. \(5(x+9) = 0\)

    • The left side is expressed as the sum of two terms

  2. \(x(x-9) - 42 = 0\)

    • The left side is a product and the right side is not 0

  3. \(x(x+3) + 6 = -2\)

    • The right side is 0

  4. \(3x = -8\)

    • The left side is 0

  5. \((x+12)(x-1) = 0\)

    • The left side involves \(x^2\)

  6. \(3x - 11 = \frac{10}{x}\)

    • One side of the equation has a term with \(3x^2\)

  7. \(4(x^2 - 1) = 8\)

    • The right side of is 0 and the left side is a product

Summary