PRESENTATION 13 Simple Equations EQUATIONS An equation is

PRESENTATION 13 Simple Equations

EQUATIONS • An equation is a mathematical statement of equality between two or more quantities • It always contains an equal sign • A formula is a particular type of equation that states a mathematical rule

WRITING EQUATIONS • The following examples illustrate writing equations from given word statements • A number plus 20 equals 35: • Let n = the number The equation would become: n + 20 = 35 • Four times a number equals 40: • • Let x = the number Four times the number would then be 4 x The equation is now: 4 x = 40

SUBTRACTION PRINCIPLE OF EQUALITY • The subtraction principle of equality states: • • If the same number is subtracted from both sides of an equation, the sides remain equal The equation remains balanced

SUBTRACTION PRINCIPLE OF EQUALITY • Procedure for solving an equation in which a number is added to the unknown: • Subtract the number that is added to the unknown from both sides of the equation

SUBTRACTION PRINCIPLE OF EQUALITY • Example: • Solve x + 5 = 12 for x: In the equation, the number 5 is added to x so subtract 5 from both sides to solve for x x + 5 = 12 – 5 x = 7

ADDITION PRINCIPLE OF EQUALITY • The addition principle of equality states: • • If the same number is added to both sides of an equation, the sides remain equal The equation remains balanced

ADDITION PRINCIPLE OF EQUALITY • Procedure for solving an equation in which a number is subtracted from the unknown • • Add the number, which is subtracted from the unknown, to both sides of an equation The equation maintains its balance

ADDITION PRINCIPLE OF EQUALITY • Example: Solve for y: • y – 7 = 10 In the equation, the number 7 is subtracted from y, so add 7 to both sides y – 7 = 10 + 7 +7 y = 17

DIVISION PRINCIPLE OF EQUALITY • The division principle of equality states: • • If both sides of an equation are divided by the same number, the sides remain equal The equation remains balanced

DIVISION PRINCIPLE OF EQUALITY • Procedure for solving an equation in which the unknown is multiplied by a number: • • Divide both sides of the equation by the number that multiplies the unknown The equation maintains its balance

DIVISION PRINCIPLE OF EQUALITY • Example: Solve for x: 6 x = 30 • In the equation, x is multiplied by 6, so divide both sides by 6 x=5

MULTIPLICATION PRINCIPLE OF EQUALITY • The multiplication principle of equality states: • • If both sides of an equation are multiplied by the same number, the sides remain equal The equation remains balanced

MULTIPLICATION PRINCIPLE OF EQUALITY • Procedure for solving an equation in which the unknown is divided by a number: • • Multiply both sides of the equation by the number that divides the unknown Equation maintains in balance

MULTIPLICATION PRINCIPLE OF EQUALITY • Example: Solve for y: • In the equation, y is divided by 3, so multiply both sides by 3 y = 15

ROOT PRINCIPLE OF EQUALITY • The root principle of equality states: • • If the same root of both sides of an equation is taken, the sides remain equal The equation remains balanced

ROOT PRINCIPLE OF EQUALITY • Procedure for solving an equation in which the unknown is raised to a power: • • Extract the root of both sides of the equation that leaves the unknown with an exponent of 1 Equation maintains in balance

ROOT PRINCIPLE OF EQUALITY • Example: Solve for x: • 2 x = 25 In the equation, x is squared, so to solve the equation, extract the square root of both sides x=5

POWER PRINCIPLE OF EQUALITY • The power principle of equality states: • • If both sides of an equation are raised to the same power, the sides remain equal The equation remains balanced

POWER PRINCIPLE OF EQUALITY • Procedure for solving an equation which contains a root of the unknown: • • Raise both sides of the equation to the power that leaves the unknown with an exponent of 1 Equation maintains in balance

POWER PRINCIPLE OF EQUALITY • Example: Solve for y: • • In the equation, y is expressed as a square root, so to solve the equation, square both sides (√y)2 = (3)2 y=9

PRACTICAL PROBLEMS • A company’s profit for the second half year is $150, 000 greater than the profit for the first half year • The total annual profit is $850, 000 • What is the profit for the first half year and the second half of the year?

PRACTICAL PROBLEMS • Let P equal the profit for the first half year • Then P + $150, 000 is the profit for the • • second half year The sum is $850, 000 Set up an equation: • P + $150, 000 = $850, 000 Sum like terms: 2 P + $150, 000 = $850, 000

PRACTICAL PROBLEMS • Use the subtraction principle of equality and subtract $150, 000 from both sides: 2 P + $150, 000 – $150, 000 = $850, 000 – $150, 000 2 P = $700, 000 • Use the division principle of equality and divide both sides by 2 2 P ÷ 2 = $700, 000 ÷ 2 P = $350, 000

PRACTICAL PROBLEMS • The profit for the first half year is $350, 000 • The profit for the second half year is $350, 000 + $150, 000 = $500, 000 • Check: Total profit is $350, 000 + $500, 000, which equals $850, 000