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Chapter 2 Linear Equations and Inequalities in One Variable § 2.1 The Addition Property of Equality The Addition Property of Equality The same real number (or algebraic expression) may be added to both sides of an equation without changing the equation’s solution. This can be expressed symbolically as follows: If a = b, then a + c = b + c Blitzer, Introductory Algebra, 5e – Slide #3 Section 2.1 Linear Equations Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the form ax + b = 0, where a and b are real numbers and a is not equal to 0. An example of a linear equation in x is 4x + 2 = 6. Linear equations in x are first degree equations in the variable x. Blitzer, Introductory Algebra, 5e – Slide #4 Section 2.1 Properties of Equality Property Definition Addition Property of Equality The same real number or algebraic expression may be added to both sides of an equation without changing the equation’s solution set. Subtraction Property of Equality The same real number or algebraic expression may be subtracted from both sides of an equation without changing the equation’s solution set. Blitzer, Introductory Algebra, 5e – Slide #5 Section 2.1 Solving Linear Equations Solving a Linear Equation 1) Simplify the algebraic expressions on each side. 2) Collect all the variable terms on one side and all the numbers, or constant terms, on the other side 3) Isolate the variable and solve. 4) Check the proposed solution in the original equation. Blitzer, Introductory Algebra, 5e – Slide #6 Section 2.1 Solving an equation using the addition property Solve the equation : x 9 12 We can isolate the variable, x, by adding 9 to both sides. x 9 12 x 9 9 12 9 x 21 The set of an equation' s solutions is called its solution set. The solution is 21 or in set notation {21}. Blitzer, Introductory Algebra, 5e – Slide #7 Section 2.1 Subtracting from both sides of an equation Since we know that subtraction is just addition of an opposite or an additive inverse, we can also subtract the same number from both sides of an equation without changing the equation’s solution. Solve: x + 6 = 9 x + 6 – 6 = 9 – 6 Subtract 6 from both sides. x=3 Blitzer, Introductory Algebra, 5e – Slide #8 Section 2.1 Adding and Subtracting Variable Terms in an Equation Now consider an equation in which we would need to subtract variable terms from both sides. Remember that our goal is to isolate all the variable terms on one side. To do this in the equation below, we must get the 4x term off the RHS by adding its opposite, -4x, to both sides. Solve: 5x = 4x + 3 5x – 4x = 4x + 3 – 4x Subtract 4x from both sides. This simplifies to: x = 3 Blitzer, Introductory Algebra, 5e – Slide #9 Section 2.1 Solving Linear Equations EXAMPLE: Solve for x 14x + 2 = 15x 2) Collect variable terms on one side and constant terms on the other side. 14x – 14x + 2 = 15x – 14x 2=x Subtract 14x from both sides Simplify Check the proposed solution of 2 in the original equation. When we insert the 2 for x, we Get the sentence 14(2) + 2 = 15(2) or we get 28 + 2 = 30, which is true. Then x = 2 is the solution. Blitzer, Introductory Algebra, 5e – Slide #10 Section 2.1 Solving Linear Equations EXAMPLE Solve and check: 5 + 3x - 4x = 1 - 2x + 12. SOLUTION 1) Simplify the algebraic expressions on each side. 5 + 3x - 4x = 1 - 2x + 12 5 - x = 13 - 2x Combine like terms: +3x - 4x = -x 1 + 12 = 13 Blitzer, Introductory Algebra, 5e – Slide #11 Section 2.1 Solving Linear Equations CONTINUED 2) Collect variable terms on one side and constant terms on the other side. 5 - x + 2x = 13 - 2x + 2x 5 + 1x = 13 5 – 5 + 1x = 13 - 5 x=8 Add 2x to both sides Simplify Subtract 5 from both sides Simplify Blitzer, Introductory Algebra, 5e – Slide #12 Section 2.1 Solving Linear Equations CONTINUED 4) Check the proposed solution in the original equation. 5 + 3x - 4x = 1 - 2x + 12 5 + 3(8) - 4(8) ?= 1 – 2(8) + 12 ? 5 +24 – 32 = 1 – 16 + 12 29 -32 ?= – 16 + 13 -3 = -3 Blitzer, Introductory Algebra, 5e – Slide #13 Section 2.1 Original equation Replace x with 8 Multiply Add or subtract from left to right True - It checks. The solution set is {8}.