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7.1Variable Notation

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In arithmetic, we perform mathematical operations with specific numbers. In algebra, we perform these same basic operations with numbers and variables- letters that stand for unknown quantities.

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Algebra is considered to be a generalization of arithmetic. In order to do algebra it is important to know the vocabulary and notation (symbols) associated with it. An algebraic expression consists of constants, variables, and operations along with grouping symbols.

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The numerical coefficient of a variable is the number that is multiplied by the variable. For example, the expression 2x + 5 has constants of 2 and 5, variable of x and x has coefficient of 2. The terms of an algebraic expression are the quantities that are added (or subtracted).

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When a term is the product of a number and letters or letters alone, no symbol for multiplication is normally shown. For example 2x means 2 times some number x and abc means some number a times some number b times some number c.

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Constants are numbers which do not change in value. Variables are unknown quantities and are represented by letters. In the expression 2x +3y -5, the 2, 3, and 5 are constants and x and y are variables.

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To evaluate an algebra expression, substitute numbers for the variables and simplify using the order of operations. It is a good idea to replace the variables with their values in parentheses.

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For example to evaluate 2x - y when x = 5 and y = -3, replace the variables with their values in parentheses 2(5) - (-3) then simplify = 13

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Terms are always separated by a plus (or minus) sign not inside parentheses. The expression 2x - 3y has two terms, 2x and -3y. 2 and -3 are constants, x and y are variables with 2 being the coefficient of x and -3 the coefficient of y. The expression 2x +3y -5 has 3 terms.

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LIKE TERMS are terms whose variable factors are the same. Like terms can be added or subtracted by adding (subtracting) the coefficients. This is sometimes referred to as combining like terms.

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Example: Simplify each expression by combining like terms. 7y - 2y 5w + w 5.1x - 3.4x 69a - 47a - 51a 2x - 6x y y -6x x -4 2x + 3y - x +9y

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If an algebraic expression that appears in parentheses cannot be simplified, then multiply each term inside the parentheses by the factor preceding the parentheses, then combine like terms.

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Example: Simplify the expression by combining like terms.

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Simplify the expression:

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If an expression inside parentheses is preceded by a “+” sign, then remove the parentheses by simply dropping them. For example: 3x + (4y + z) = 3x + 4y + z

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If an expression in parentheses is preceded by a “-” sign then it is removed by changing the sign of each term inside the parentheses and dropping the parentheses. 3x – (4y – z) = 3x – 4y + z

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Example: Simplify the expression by combining like terms.

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An equation is a statement that 2 expressions are equal. The symbol “=“ is read “is equal to” and divides the equation into 2 parts, the left member and the right member. In the equation 2x + 3 = 13, 2x + 3 is the left member and 13 is the right member.

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The solution to an equation in one variable is the number that can be substituted in place of the variable and makes the equation true. For example 5 is a solution to the equation 2x + 3 = 13 because 2(5) + 3 = 13 is true.

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To solve an equation means to find all solutions or roots for the equation.

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Solve each equation: z = p = 3(9) – 5 b = 5(3) – 4(8) + 7

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To write a verbal statement into a symbolic statement: Assign a letter to represent the missing number. Identify key words or phrases that imply or suggest specific mathematical operations. Translate words into symbols.

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Write the statements into symbols: 8 more than a number is less than 3 times a number is 45. The sum of 15, 4 and a third number is zero. 8 + n = 34 3x – 5 = t = 0

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