2.3 Solve Two-Step Equations Essential question: How do you solve two step equations? Warm-up: Solve the equation. 1.3x = b + 21 = 11 3.Simplify the expression. 3(x + 2) 4x There are three times as many goats as sheep in a petting zoo. Find the total number sheep if the total number of goats and sheep is 28. CommonCoreCC.9-12.A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters EXAMPLE 1 Solve a two-step equation Solve + 5 = 11. x 2 Write original equation. + 5 = x 5 = x 2 11 5 Subtract 5 from each side. = x 2 6 Simplify. 2 = x Multiply each side by 2. x = 12 Simplify. ANSWER The solution is 12. Check by substituting 12 for x in the original equation. EXAMPLE 1 Solve a two-step equation CHECK + 5 = x 2 11 Write original equation = 12 2 ? Substitute 12 for x. Simplify. Solution checks. 11 = 11 GUIDED PRACTICE for Example 1 Solve the equation. Check your solution. 1. 5x + 9 = 24 3 ANSWER GUIDED PRACTICE for Example 1 Solve the equation. Check your solution. 2. 4y 4 = 16 5 ANSWER GUIDED PRACTICE for Example 1 Solve the equation. Check your solution. 3. 1 = 7 z 3 18 ANSWER Solve a two-step equation by combining like terms EXAMPLE 2 Solve 7x 4x = 21 7x 4x = 21 Write original equation. 3x = 21 Combine like terms. Divide each side by 3. x = 7 Simplify. = 3 3x3x 3 21 Find an input of a function EXAMPLE 3 The output of a function is 3 less than 5 times the input. Find the input when the output is 17. y = 5x 3 y is 3 less than 5 times x. SOLUTION Write an equation for the function. Let x be the input and y be the output STEP 1 Solve the equation for x when y = 17. STEP 2 Find an input of a function EXAMPLE 3 y = 5x 3 Write original function. 17 = 5x 3 Substitute 17 for y = 5x Add 3 to each side. 20 = 5x Simplify. Divide each side by 5. 4 = x Simplify. ANSWER An input of 4 produces an output of 17. = x5x 5 Find an input of a function EXAMPLE 3 Write original function. Substitute 17 for y and 4 for x. Multiply 5 and 4. Simplify. Solution checks. CHECK 17 = 20 3 ? 17 = 5(4) 3 ? 17 = 17 y = 5x 3 GUIDED PRACTICE for Examples 2 and w + 2w = 24 Solve the equation. Check your solution. 4 ANSWER GUIDED PRACTICE for Examples 2 and t 3t = 35 Solve the equation. Check your solution. 7 ANSWER GUIDED PRACTICE for Examples 2 and 3 6. 16 = 5d 9d Solve the equation. Check your solution. 4 ANSWER GUIDED PRACTICE for Examples 2 and 3 7. The output of a function is 5 more than 2 times the input. Find the input when the output is 11. ANSWER 3 GUIDED PRACTICE for Examples 2 And 3 8. The output of a function is 4 less than 4 times the input. Find the input when the output is 3 ANSWER 3 4 1 EXAMPLE 4 Solve a multi-step problem As a scuba diver descends into deeper water, the pressure of the water on the divers body steadily increases. SCUBA DIVING The pressure at the surface of the water is 2117 pounds per square foot ( lb/ft 2 ). The pressure increases at a rate of 64 pounds per square foot for each foot the diver descends. Find the depth at which a diver experiences a pressure of 8517 pounds per square foot. SOLUTION EXAMPLE 1 Write a verbal model. Then write an equation. Find the depth at which the pressure is 8517 pounds per square foot. P = d Write equation. STEP 1 STEP 2 EXAMPLE 4 Solve a multi-step problem GUIDED PRACTICE Substitute 8517 for P 2117 = 2117 d Subtract 2117 from each side = d 6400 = 64d Simplify. Divide each side by = d Simplify. ANSWER A diver experiences a pressure of 8517 pounds per square foot at a depth of 100 feet = 64d EXAMPLE 4 Solve a multi-step problem GUIDED PRACTICE P = d Write original equation. Substitute 8517 for P and 100 for d. Multiply 64 and 100. Simplify. Solution checks ? = (100) ? = CHECK EXAMPLE 4 Solve a multi-step problem 8517 = 8517 GUIDED PRACTICE for Example 4 WHAT IF? In Example 4, suppose the diver experiences a pressure of 5317 pounds per square foot. Find the divers depth. 9. ANSWER 50 ft GUIDED PRACTICE for Example 4 Jobs Kim has a job where she makes $8 per hour plus tips. Yesterday, Kim made $53 dollars, $13 of which was from tips. How many hours did she work? h ANSWER Classwork/Homework 2.3 Exercises 2-44 even pages 94-96