Solving Two-Step Equations Section 2-2

Goals Goal To solve two-step equations in one variable. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

This equation contains multiplication and addition. Equations that contain two operations require two steps to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse operations to undo them in reverse over one at a time. Many equations contain more than one operation, such as 2x + 5 = 11. Operations in the equation  multiplied First x is multiplied by 2.  added. Then 5 is added.   To solveSubtract 5 from both sides of the equation. divide both sides by 2. Then divide both sides by 2. Solving Two-Step Equations

2x + 5 =11 –5 2x = 6 x = 3 Subtract 5 from both sides of the equation. Divide both sides of the equation by 2. The solution set is {3}. Each time you perform an inverse operation, you create an equation that is equivalent to the original equation. Equivalent equations have the same solutions, or the same solution set. In the example above, 2x + 5 = 11, 2x = 6, and x = 3 are all equivalent equations.

To Solve: Inverse Operations in the Inverse Order Ex: x + 9 = 6 5 Ask yourself: What is the first thing we are doing to x? The second thing? Recall the order of operations as you answer these questions. dividing by 5 adding 9 To undo these steps, do the inverse operations in inverse order.

The DO-UNDO Chart Use a chart as a shortcut to answering the questions. DO UNDO ÷ ·5 Follow the steps in the ‘undo’ column to isolate the variable. Ex: x + 9 = 6 5 First subtract 9. x = x = -3 5 Then multiply by 5. (5) x = -3(5) 5 x = -15

Example: Complete the do-undo chart. DO UNDO -2 ·3 ÷ 3 +2 To solve for d: First multiply by 3. Then add 2. Ex: d - 2 = 7 3 (3) d - 2 = 7(3) 3 d - 2 = d = 23

Example: Remember to always use the sign in front of the number. DO UNDO ÷ · -7 To solve for a: First subtract 3. Then multiply by -7. Ex: 3 - a = a = a = -5 7 (-7)(- a) = (-5)(-7) 7 a = 35

Your Turn: 1)5z + 16 = 51 2)14n - 8 = 34 3)4b + 8 = 10 -2

The answers: 1)DO UNDO · ÷ 5 1)z = 7 2) DO UNDO · ÷ 14 2)n = 3 3)DO UNDO · 4 · ÷ -2 ÷ 4 3)b = -7

Solve the equation. Check your answer. 7x = 7 –4 + 7x = 3 Add 4 to both sides. 7x = 7 is equivalent to –4 + 7x = 3. Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. The solution set is {1}. First x is multiplied by 7. Then –4 is added. –4 + 7x = x = 1 Your Turn:

Check –4 + 7(1) 3 3 To check your solution, substitute 1 for x in the original equation. Check your answer. –4 + 7x = 3 – Your Turn: Continued

Solve the equation. 7.2 = 1.2y 1.5 = 1.2y – 5.7 Add 5.7 to both sides. 7.2 = 1.2y is equivalent to 1.5 = 1.2y – 5.7. Since y is multiplied by 1.2, divide both sides by 1.2 to undo the multiplication. The solution set is {6}. First y is multiplied by 1.2. Then 5.7 is subtracted = y 1.5 = 1.2y – 5.7 Your Turn:

Solve the equation. n = 0 Subtract 2 from each side. Since n is divided by 7, multiply both sides by 7 to undo the division. The solution set is {0}. First n is divided by 7. Then 2 is added. –2 = 0 = 0 is equivalent to + 2 = 2. Your Turn:

Solve the equation. Method 1 Use fraction operations. Since is subtracted from, add to both sides to undo the subtraction. Example: Two-Step Equations with Fractions

y = 16 Since y is divided by 8 multiply both sides by 8. Simplify. The solution set is {16}. Example: Continued

Method 2 Multiply by the least common denominator (LCD) to clear fractions. y – 6 = y = 16 Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the left side. Simplify. Since 6 is subtracted from y, add 6 to both sides to undo the subtraction. The solution set is {16}. Example: Continued

Solve the equation. Method 1 Use fraction operations. Since is added to, subtract from both sides to undo the addition. Example: Two-Step Equations with Fractions

Since r is multiplied by multiply both sides by, the reciprocal. Simplify. The solution set is. Example: Continued

Method 2 Multiply by the least common denominator (LCD) to clear the fractions. Multiply both sides by 12, the LCD of the fractions. Distribute 12 on the left side. Example: Continued

8r + 9 = 7 – 9 –9 8r =–2 Simplify. Since 9 is added 8r, subtract 9 from both sides to undo the addition. Since r is multiplied by 8, divide both sides 8 to undo the multiplication. The solution set is. Example: Continued

You can multiply both sides of the equation by any common denominator of the fractions. Using the LCD is the most efficient. Helpful Hint

Solve the equation. Check your answer. Method 1 Use fraction operations. Since is subtracted from, add to both sides to undo the subtraction. Your Turn:

Simplify. The solution set is. Since x is multiplied by multiply both sides by, the reciprocal. Your Turn: Continued

Method 2 Multiply by the least common denominator (LCD) to clear the fractions. Multiply both sides by 10, the LCD of the fractions. Distribute 10 on the left side. 4x – 5 = 50 Your Turn: Continued

x = 55 Simplify. Since 5 is subtracted from 4x add 5 to both sides to undo the subtraction. Simplify. Since x is multiplied by 4, divide both sides 4 to undo the multiplication. 4x – 5 = 50 The solution set is. Your Turn: Continued

Solve the equation. Method 1 Use fraction operations. Since is added to, subtract from both sides to undo the addition. Your Turn:

Simplify. Since u is multiplied by multiply both sides by the reciprocal,. The solution set is. Your Turn: Continued

Method 2 Multiply by the least common denominator (LCD) to clear fractions. Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the left side. 6u + 4 = 7 Your Turn: Continued

– 4 6u = 3 Simplify. Since 4 is added to 6u subtract 4 from both sides to undo the addition. Simplify. Since u is multiplied by 6, divide both sides 6 to undo the multiplication. 6u + 4 = 7 The solution set is. Your Turn: Continued

Solve the equation. Method 1 Use fraction operations. Since is subtracted from, add to both sides to undo the subtraction. Your Turn:

Simplify. The solution set is {15}. n = 15 Since n is multiplied by multiply both sides by the reciprocal,. Your Turn: Continued

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Assignment 2.2 Exercises Pg. 98 – 100: #10 – 68 even