Fractions and Linear Equations


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1 Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps for full credit, but you are encouraged to check your answers using the calculator. Reducing: A fraction is reduced if the numerator and denominator have no common factors. You can reduce a fraction by dividing both numerator and denominator by the same value until they have no factors in common. Example: Consider the fraction 18/1. Dividing both 18 and 1 by the common factor results in the fraction 9/6. Dividing both 9 and 6 by the common factor results in the fraction /. Because the values and have no factors in common, the fraction has been reduced to lowest terms: 18 1 = 9 6 = Addition/Subtraction: To add or subtract fractions, they must have the same denominator. If two fractions do not have a common denominator, this is achieved by multiplying both numerator and denominator by the same number, which does not change the value of the fraction. Example: To add the fractions 1/ and 1/5, we must first find a common multiple of the denominators and 5. The smallest, or least, common multiple is the value. To obtain a common denominator, both numerator and denominator of 1/ are multiplied by 5 and both numerator and denominator of 1/5 are multiplied by. The fractions are then added by combining the numerators. A pictorial representation of the sum is also provided below = 5 + = 7 Subtraction is similar: = 5 = To show your work in computing the sum 1/ + 1/5 online, type 1/ + 1/5 = 5/ + / = 7/. If you include an additional step, keep in mind the proper use of parentheses: 1/ + 1/5 = 5/ + / = (5 + )/ = 7/. The expression 5 + / would be 5 + = 6 5.
2 Multiplication: To multiply two fractions, the numerators are multiplied together and the denominators are multiplied together. Keep in mind that the denominator of an integer is an understood 1. Examples: 5 = 5 = = 1 7 = 1 7 = 1 7 = 9 Division: In order to divide by a fraction, multiply by its reciprocal. That is, reverse the numerator and denominator of the fraction by which you are dividing and change the operation from division to multiplication. Examples: 5 = 5 = 1 = = 1 7 = 81 Linear Equations At this time only equations in one variable will be addressed. A linear equation is one in which the variable is raised only to the first power. To solve a linear equation, isolate the variable on one side of the equation by performing the same operation (addition, subtraction, multiplication, or division) to both sides of the equation. You have likely been solving linear equations since grade school, where you were asked to write into the box the missing number: + = 5. This is no different than solving the linear equation + x = 5. When dealing with variable expressions, remember that you can only combine like terms, which contain the same variable(s) raised to the same powers. For example, x + x = 5x, but + x cannot be simplified further. Example 1: To solve the equation = 9x 1x for x, first combine the like terms on the righthand side of the equation and then divide both sides by  to isolate the x. = 9x 1x = x = x = x When dealing with negative fractions, the negative sign can be placed in front of the fraction or either in the numerator or denominator: = =. You can verify that your answer is correct by replacing the variable in the original equation with the value found: = 9(/) 1(/) = Because this is a true equation, the solution x = / is correct.
3 Example : To solve the equation below, we can start by multiplying all terms on both sides by the denominator to eliminate the fraction. 9x = 11x ( 9x ) = 11x 9x = 8 x 9x + x = 8 x + x 1x = 8 1x 1 = 8 1 x = 8 1 Example : If two fractions are equal to each other, then you can cross multiply to eliminate the fractions. That is, multiply the denominator of each fraction by the numerator of the other and set the products equal. In the solution of the equation below, this is equivalent to multiplying both sides by the common denominator, 18. To show your work in solving the above equation online, you could type: x(9) = (x 5) 7x = 8x 19x =  x = /19 Example : If either side of the equation contains another term, cross multiplication cannot be used. In the following example, the fractions are eliminated by multiplying through by the common denominator, 7. b = b 7 7 b + 7 ( ) = 7 ( b 7 7 ) b = b 17 = b
4 Reduce each fraction to lowest terms Fractions and Linear Equations Examples (Solutions provided at bottom of page) Perform each fraction operation by hand. Give the answer in fractional form reduced to lowest terms Solve each equation. Give noninteger solutions in fractional form reduced to lowest terms = 6 + x 9. 1(x + ) + 1 = 5x . 1 = x x 5 + = (x ) 1. x+8 5 = 7x (x+7) 1 = (x 11) (x + ) + 1 = 8x x + = (x 6) Solutions: 1. 11/1. /. 6/11. / + / = 1/ 5. / / = 7/ 6. 15/8 7. / * /5 = 6/0 = / 8. x = 8 9. x = 51/7. x = x = 5/8 1. x = 6/7 1. x = 11/7 1. x = x = 5/16
5 Worksheet 5 Fractions and Linear Equations 1. Reduce each fraction to lowest terms a. b. 90 c. d. 5 6 e. 5 0 Perform each fraction operation by hand. You must show your work for full credit. Give the answer in fractional form reduced to lowest terms Solve each equation. Give noninteger solutions in fractional form reduced to lowest terms. 6. x 9 = 7. 8x 5 = 9 + (x + 7) 8. 8(x + 1) + x = 9 9. x+9 = 11x 5. 5(x+) = 8(x )
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