Chapter 2. Functions, Linear equations, and inequalities. Solutions

Section 2.1: Linear equations Chapter 2: Functions, Linear equations, and inequalities Solving Linear Equations Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. For example: Solve, be sure to check your answer. (These problems only require step 3 and step 5, or step 4 and step 5. Neither of these problems require steps 1 or 2.) Solve: 𝑥 – 7 = 13 Requires steps 3 & 5 Solve: 2𝑥 = 12 Requires steps 4 & 5 Step 3: 𝑥 − 7 = 13 +7 + 7 Answer: 𝑥 = 20 Step 4: Step 5: Check, replace 𝑥 with 20 in original problem and confirm the sides of the equation are equal after simplifying. 20 – 7 = 13 13 = 13 ✓ 2𝑥 2 12 = 2 Answer: 𝑥 = 6 Step 5: Check 2(6) = 12 12 = 12 ✓ For example: Solve, be sure to check your answer. (This problem will require steps 3, 4 and 5. It will not require step 1 nor 2.) Solve: 3x – 8 = 22 Step 3: 3𝑥 − 8 = 22 +8 + 8 3x = 30 For example: Solve, be sure to check your answer. (This problem will require all 5 steps.) Solve: 2𝑥 + 3(8𝑥 − 14) = 4 + 3𝑥 Step 1: First clear parenthesis: 2𝑥 + 24𝑥 – 42 = 4 + 3𝑥 Next combine 2𝑥 + 24𝑥 Step 4: 26𝑥 – 42 = 4 + 3𝑥 3𝑥 3 Step 2: Get the x-terms on the left side. = 30 3 Answer: 𝑥 = 10 Step 5: Check 3(10) – 8 = 22 30 – 8 = 22 22 = 22 ✓ 26𝑥 – 42 = 4 + 3𝑥 −3𝑥 − 3𝑥 23𝑥 – 42 = 4 Step 3: Get the numbers without the variables on the right side. 23𝑥 – 42 = 4 +42 + 42 23𝑥 = 46 Step 4: 23𝑥 46 = 23 23 Answer: 𝑥 = 2 Step 5: Check 2(2) + 3(8(2) − 14) = 4 + 3(2) 2(2) + 3(2) = 4 + 3(2) 4 + 6 = 4 + 6 10 = 10 ✓ Section 2.1: Linear equations Chapter 2: Functions, Linear equations, and inequalities #1 – 20: Solve, be sure to check your answer. (These problems only require step 3 and step 5, or step 4 and step 5. None of these problems require steps 1 and 2.) Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 1) 𝑥 + 5 = 10 2) 𝑦 + 6 = 14 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 3) 𝑥 − 13 = 16 4) y – 15 = 23 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 1 5 2 2 3 11 4 4 5) 𝑥 + = 6) 𝑦 + = Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 7) 𝑦 − 3 = 8) 𝑧 − 5 = 1 2 2 3 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 9) 3𝑥 = 21 10) 4𝑦 = 28 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 11) −6𝑧 = 30 12) −4𝑥 = 48 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 13) 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 14) 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 2 3 3 4 𝑥 = 18 𝑦 = 21 #21 -44: Solve, be sure to check your answer. (These problems will require most of steps 2, 3, 4 and 5. They will not require step 1.) Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 21) 6𝑥 + 10 = 22 22) 5𝑥 + 18 = 38 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 23) 8𝑥 – 19 = 13 24) 6𝑧 – 20 = 40 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 25) 26) 2 3 3 4 𝑥 + 24 = 30 𝑦 + 15 = 21 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 27) 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 28) 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. −2 3 −3 2 𝑦 − 14 = −28 𝑧 − 5 = −11 Section 2.1: Linear equations Chapter 2: Functions, Linear equations, and inequalities #45 – 64: Solve, be sure to check your answer. (These problems will require all 5 steps.) Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 45) 3𝑥 + 55 − 9𝑥 = 24 + 4𝑥 − 9 46) 19𝑥 + 14 − 6𝑥 = 32 + 8𝑥 − 8 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 47) 3𝑥 + 55 + 7 = 24𝑥 + 4𝑥 + 12 48) 9𝑥 + 14 − 2 = −32 + 7𝑥 − 8 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 49) 3(5𝑦 − 4) = 48 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 50) 2(3𝑦 − 11) = 8 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 51) −3(2𝑥 − 5) = −9 52) −7(4𝑥 − 1) = −49 Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 53) 21 = −7(2𝑥 − 3) 54) 15 = −2(5𝑥 − 1) Steps to solve a linear equation: 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 55) 5𝑥 + 3(6𝑥 − 7) = 25 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. • Multiply by the reciprocal if the coefficient of the variable is a fraction. 5) Check the solution. • Substitute the solution into the original equation to make sure the result is a true statement. 56) 2𝑥 + 4(3𝑥 − 5) = 22