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2º ESO BIL Dpto. de Matemáticas- I.E.S. Montes Orientales (Iznalloz)- Nieves Fiestas Carmona. Curso 2011/2012 UNIT 6. EQUATIONS EQUATIONS AND THEIR COMPONENTS What is an equation? Components An equation is an expression stating the equality of tow algebraic expressions. This equality exists only for certain values of the letters. Ex: x + 6 = 2x + 1, is true for x = 5 ( 5 + 6 = 2 · 5 + 1 ) We call each side of the equal sign, expression, the letters are called unknowns, the addends are called terms and the values for which the equation is true are called solutions. Ex: x + 6 = 2x + 1 First expression: x + 6. Second expression: 2x + 1. Unknown: x. Terms: x, 6, 2x, 1. Solution: x = 5. BASIC EQUATION-SOLVING TECHNIQUES Adding or Subtracting from both expressions If the same amount is added or subtracted from both expressions in an equation, the result is another, equivalent equation (with the same solution). Rule: A number that is added /subtracted in one expression can be eliminated from that expression if it is subtracted /added from the other expression. Ex 1: x + 2 = 5 -( x + 2 – 2 = 5 – 2) - x = 5 – 2 - x = 3 Ex 2: x - 1 = 3 -( x - 1 + 1 = 3 + 1) - x = 3 + 1 - x = 4 Dividing or Multiplying both expressions If both expressions in an equation are divided or multiplied by the same number, the result is another, equivalent equation. Rule: A multiplication / division operation can be eliminated from one expression if it is added as a division / multiplication operation to the other expression. Ex 1: 2x = 6 -( 2x / 2 = 6 / 2) - x = 6 / 2 - x = 3 Ex 2: x 4 3 x 3 (4) 3 - x = - 12 3 FIRST DEGREE EQUATIONS A first-degree equation is an equation than can be reduced to the form ax=b. It has a single solution of x = b/a. Solving equations To solve a first degree equation with one unknown, you reduce the expressions and move the terms until the unknown is isolated. Ex: 6x - 4 = 3x + 6 6x – 3x = 6 + 4 First, we move the terms 3x = 10 Second, we reduce the terms adding x= 10 3 Third, we work out the unknown SOLVING PROBLEMS Equations as tools b) Equations can help you to solve problems. In each of the sample equations, study the steps that must be followed. Ex: A number and the same number plus one add up to 43. What are the two numbers that add up to 43? a) Identify the known and unknown numbers: A number The number plus one The sum of both n n+1 43 Find the relationship between the variables. A number + The same number plus one = 43 Equation: n c) + (n + 1) = 43 Solve the equation. n + n + 1 = 43 2n = 43 – 1 2n = 42 n = 21 d) Write the solution: n = 21 ; n + 1 = 21 + 1 = 22 The numbers are 21 and 22