Presentation on theme: "Daily Quiz - Simplify the expression, then create your own realistic scenario for the final expression."— Presentation transcript:
Daily Quiz - Simplify the expression, then create your own realistic scenario for the final expression.
Simplify Expression Check Complete in your notes as Practice! 1. 2. 3. Multiply the quantity by (-5) and add the product to the quantity
Objectives: SWBAT… Create and carry out a plan for solving equations Maintain equality when solving equations through inverse operations and simplification techniques (such as combining like terms) Solve one-step linear equations Solve multi-step linear equations
●A variable is a letter which represents an unknown number. Any letter can be used as a variable. ●An algebraic expression contains at least one variable. Examples: a, x+5, 3y – 2z ●A verbal expression uses words to translate algebraic expressions. Example:“The sum of a number and 3” represents “ n+3. ” ●An equation is a sentence that states that two mathematical expressions are equal. ●Linear Equation in One Variable - can be written in the form ax +b =c, a 0 Example: 2x-16=18 Review of Key Concepts
Key Concepts Continued ●To solve means to find the value of a variable ●Inverse Operations are operations that “undo” each other ●division and multiplication ●addition and subtraction ●Isolate a Variable is part of the process of solving, in which the variable is placed on one side of the equation by itself ●Equality is the state of being equal or having the same value – we always maintain equality when solving equations ●A solution is a value that can take the place of a variable to make an equation true
Solving equations is just a matter of undoing operations that are being done to the variable. In a simple equation, this may mean that we only have to undo one operation, as in the following example. Solve the following equation for x x + 3 = 8 x + 3 = 8 the variable is x x + 3 – 3 = 8 – 3 we are adding 3 to the variable, so to get rid of the added 3, we do the opposite--- subtract 3. x = 5 remember to do this to both sides of the equation. Single-Step Linear Equation
In an equation which has more than one operation, we have to undo the operations in the correct order. Solve the following equation: 5x – 2 =13 5x – 2 = 13 The variable is x 5x – 2 + 2 = 13 + 2 We are multiplying it by 5, and subtracting 2 First, undo the subtracting by adding 2. 5x = 15 Then, undo the multiplication by dividing by 5. 5 5 x = 3 Multi-Step Linear Equation We start with the operation the farthest away from the variable!
Steps to Solving Equations ●Simplify each side of the equation, if needed, by distributing or combining like terms. ●Move variables to one side of the equation by using the opposite operation of addition or subtraction. ●Isolate the variable by applying the opposite operation to each side. First, use the opposite operation of addition or subtraction. Second, use the opposite operation of multiplication or division. ●Check your answer.
How can we “undo” operations? Isn’t this wrong? Addition Property of Equality – states you can add the same amount to both sides of an equation and the equation remains true. 2 + 3 = 5 2 + 3 + 4 = 5 + 4 9 = 9 ? true Subtraction Property of Equality – states you can subtract the same amount from both sides of an equation and the equation remains true. 4 + 7 = 11 4 + 7 – 3 = 11 – 3 8 = 8 ? true
Example 5(3 + z) – (8z + 9) = – 4z 15 + 5z – 8z – 9 = – 4z (Use distributive property) 6 – 3z = – 4z (Simplify left side) 6 + z = 0 (Simplify both sides) z = – 6 (Simplify both sides) 6 – 3z + 4z = – 4z + 4z (Add 4z to both sides) 6 + ( – 6) + z = 0 +( – 6) (Add –6 to both sides)
Multiplication Property of Equality – states you can multiply the same amount on both sides of an equation and the equation remains true. 4 · 3 = 12 2 · 4 · 3 = 12 · 2 24 = 24 Division Property of Equality – states you can divide the same amount on both sides of an equation and the equation remains true. 4 · 3 = 12 2 12 = 6 2
Example – y = 8 y = – 8(Simplify both sides) ( – 1)( – y) = 8( – 1) (Multiply both sides by –1)
Example Recall that multiplying by a number is equivalent to dividing by its reciprocal 3z – 1 = 26 3z = 27 (Simplify both sides) z = 9 (Simplify both sides) 3z – 1 + 1 = 26 + 1 (Add 1 to both sides) (Divide both sides by 3)
Special Cases No Solution – we arrive at an answer that does not maintain equality Infinite – we arrive at an answer that will always maintain equality (always be true)