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