Transcript Algebra
Chapter M3 Algebra Learning Objectives • • • • Understand and use algebraic terms Manipulate algebraic expressions Solve simple linear equations (including transposition) Solve simultaneous linear equations (including the graphical technique) • Solve business problems using simple algebra © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 1 Definitions Algebra applies quantitative concepts to unknown quantities represented by symbols. A term is a part of an expression that is connected to another term by a plus or minus sign. A constant is a term whose value does not change. A variable is a term that represents a quantity that may have different values. An expression is a combination of constants and variables using arithmetic operations. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 2 Definitions A coefficient is a factor by which the rest of a term is multiplied. The degree of expression is the highest exponent of any variable in the expression. An equation is a statement that two expressions are equal. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 3 Algebraic rules Rule: If an expression contains like terms, these terms may be combined into a single term. Like terms are terms that differ only in their numerical coefficient. Constants may also be combined into a single constant. Example: 5 x and 3 x are like terms 5 x and - 3 x may be combined 5 x - 3x 2 x © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 4 Algebraic rules Rule: When an expression is contained in brackets, each term within the brackets is multiplied by any coefficient outside the brackets. Example: Consider the expression : 23 x 4 y 1 to remove the brackets 23 x 24 y 2 1 6 x 8 y 2 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 5 Algebraic rules Rule: To multiply one expression by another, multiply each term of one expression by each term of the other expression. The resulting expression is said to be the product of the two expressions. The product of the two expressions Example: 3 x 2 and 2 x 1 3 x 2 2 x 1 3 x 2 x 1 22 x 1 6x2 3x 4x 2 6x2 x 2 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 6 Algebraic rules Rule: Any term may be transposed from one side of an equation to the other. When the transposition is made, the sign of the term must change from its original sign. ‘+’ becomes a ‘-’ and ‘-’ becomes a ‘+’. Example: 15x - 20 = 12 - 4x 15x - 20 + 4x = 12 15x + 4x = 12 + 20 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 7 Solving linear equations Solve 1. 9x - 27 = 4x + 3 for x Place like terms of the variable on the left side of the equation and the constant terms on the right side. 9x - 4x = 3 + 27 2. Collect like terms and constant terms. 5x = 30 3. Divide both sides of the equation by the coefficient of the variable (in this case 5). x=6 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 8 Solving simultaneous equations (finding the value of two variables) Solve for x and y 3x + 4y = 33 (1) 2x - 3y = 5 (2) Step 1: Make the coefficient of either of the variables in one equation equal to its coefficient in the other equation. Multiply both sides of equation (1) by 2 and equation (2) by 3. 6x + 8y = 66 (3) 6x - 9y = 15 (4) © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 9 Solve for x and y: 3x + 4y = 33 (1) 2x - 3y = 5 (2) Step 2: Eliminate the variable that has the same coefficient by: subtracting equation (4) from equation (3). If the signs are the same, then subtract equations. If the signs are different, then add equations. 6x + 8y = 66 (3) minus:6x - 9y = 15 (4) equals 8y-(-9y) = 51 17y = 51 Divide both sides by 17 to find y. y = 3 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 10 Solve for x and y: 3x + 4y = 33 (1) 2x - 3y = 5 (2) Substitute in 3 for y in equation (1) and solve for x. 3x + 4(3) = 33 3x + 12 = 33 3x = 33 - 12 3x = 21 Divide both sides by 3. x =7 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 11