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Solving Equations

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This document provides an overview of equations in one variable, including: - Defining equations and expressions, and distinguishing between the two - Identifying linear equations and determining if a number is a solution - Explaining properties of equality like addition, subtraction, multiplication, and division - Outlining the steps to solve linear equations in one variable - Describing types of linear equations like conditional, identity, and contradiction

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Equations in One Variable • Lesson 4.1 – Page 89-96 Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Equations and inequalities compare algebraic expressions. An equation is a statement that two algebraic expressions are equal. An equation always contains an equals symbol, while an expression does not. 3x – 7 = 2 3x – 7 Left side Right side Equation (to solve) Copyright © 2012, 2008, 2004 Pearson Education, Inc. Expression (to simplify or evaluate) Slide 2.1- 2 Distinguish between expressions and equations.
Decide whether each of the following is an equation or an expression. 9x + 10 = 0 9x + 10 equation expression Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 2.1- 3 CLASSROOM EXAMPLE 1 Distinguishing between Expressions and Equations Solution:
Linear Equation in One Variable A linear equation in one variable can be written in the form Copyright © 2012, 2008, 2004 Pearson Education, Inc. Ax + B = C, where A, B, and C are real numbers, with A ¹ 0. A linear equation is a first-degree equation, since the greatest power on the variable is 1. Slide 2.1- 4 Identify linear equations, and decide whether a number is a solution of a linear equation.
More Properties of Equality If the operation done to one side is also done to the other then the value of the equation does not change PAGE 89 Definition Examples • Addition: If a=b, then a + c = b + c • Subtraction: If a=b, then a – c = b – c Multiplication: If a=b, then a ∙ c = b ∙ c • Division: If a = b, then a / c = b / c (c≠0) Copyright © 2012, 2008, 2004 Pearson Education, Inc. • If x = 12, then x + 3 = 12 + 3 • If x = 12, then x – 3 = 12 – 3 • If x = 12, then x ∙ 3 = 12 ∙ 3 • If x = 12, then x / 3 = 12 / 3
CLASSROOM EXAMPLE 2 Using the Properties of Equality to Solve a Linear Equation Solve. 4x + 8x = –9 + 17x – 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solving a Linear Equation in One Variable Step 1 Clear fractions or decimals. Eliminate fractions by multiplying each side by the least common denominator. Eliminate decimals by multiplying by a power of 10. Step 2 Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed. Step 3 Isolate the variable terms on one side. Use the addition property to get all terms with variables on one side of the equation and all numbers on the other. Step 4 Isolate the variable. Use the multiplication property to get an equation with just the variable (with coefficient 1) on one side. Step 5 Check. Substitute the proposed solution into the original equation. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 2.1- 7 Solve linear equations by using the addition and multiplication properties of equality.
Type of Linear Equation Number of Solutions Copyright © 2012, 2008, 2004 Pearson Education, Inc. Indication when Solving Conditional One Final line is x = a number. Identity Infinite; solution set {all real numbers} Final line is true, such as 0 = 0. Contradiction None; solution set Æ Final line is false, such as –15 = –20 . Slide 2.1- 8 Identify conditional equations, contradictions, and identities.

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Solving Equations

  • 1. Equations in One Variable • Lesson 4.1 – Page 89-96 Copyright © 2012, 2008, 2004 Pearson Education, Inc.
  • 2. Equations and inequalities compare algebraic expressions. An equation is a statement that two algebraic expressions are equal. An equation always contains an equals symbol, while an expression does not. 3x – 7 = 2 3x – 7 Left side Right side Equation (to solve) Copyright © 2012, 2008, 2004 Pearson Education, Inc. Expression (to simplify or evaluate) Slide 2.1- 2 Distinguish between expressions and equations.
  • 3. Decide whether each of the following is an equation or an expression. 9x + 10 = 0 9x + 10 equation expression Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 2.1- 3 CLASSROOM EXAMPLE 1 Distinguishing between Expressions and Equations Solution:
  • 4. Linear Equation in One Variable A linear equation in one variable can be written in the form Copyright © 2012, 2008, 2004 Pearson Education, Inc. Ax + B = C, where A, B, and C are real numbers, with A ¹ 0. A linear equation is a first-degree equation, since the greatest power on the variable is 1. Slide 2.1- 4 Identify linear equations, and decide whether a number is a solution of a linear equation.
  • 5. More Properties of Equality If the operation done to one side is also done to the other then the value of the equation does not change PAGE 89 Definition Examples • Addition: If a=b, then a + c = b + c • Subtraction: If a=b, then a – c = b – c Multiplication: If a=b, then a ∙ c = b ∙ c • Division: If a = b, then a / c = b / c (c≠0) Copyright © 2012, 2008, 2004 Pearson Education, Inc. • If x = 12, then x + 3 = 12 + 3 • If x = 12, then x – 3 = 12 – 3 • If x = 12, then x ∙ 3 = 12 ∙ 3 • If x = 12, then x / 3 = 12 / 3
  • 6. CLASSROOM EXAMPLE 2 Using the Properties of Equality to Solve a Linear Equation Solve. 4x + 8x = –9 + 17x – 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc.
  • 7. Solving a Linear Equation in One Variable Step 1 Clear fractions or decimals. Eliminate fractions by multiplying each side by the least common denominator. Eliminate decimals by multiplying by a power of 10. Step 2 Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed. Step 3 Isolate the variable terms on one side. Use the addition property to get all terms with variables on one side of the equation and all numbers on the other. Step 4 Isolate the variable. Use the multiplication property to get an equation with just the variable (with coefficient 1) on one side. Step 5 Check. Substitute the proposed solution into the original equation. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 2.1- 7 Solve linear equations by using the addition and multiplication properties of equality.
  • 8. Type of Linear Equation Number of Solutions Copyright © 2012, 2008, 2004 Pearson Education, Inc. Indication when Solving Conditional One Final line is x = a number. Identity Infinite; solution set {all real numbers} Final line is true, such as 0 = 0. Contradiction None; solution set Æ Final line is false, such as –15 = –20 . Slide 2.1- 8 Identify conditional equations, contradictions, and identities.