Equations Solve equations using multiplication or division. Quantitative Skills Solve equations using addition or subtraction. Topic 1-C Solve equations using more than one operation. Equations Solve equations containing multiple unknown terms. Solve equations containing parentheses. Solve equations that are proportions. Solve Equations Using Multiplication or Division Key Terms The letters (x,y,z) represent unknown amounts andAn equation is a mathematical statement in which two are called unknowns or variables.quantities are equal. The numbers are called known or given amounts.Solving an equation means finding the value of anunknown.For example: 8x = 24 4x = 16To solve this equation, the value of x must bediscovered.Division is used to solve this equation. How to solve an equation withRemember! multiplication and division 8x = 24Any operation performed on one side of the equationmust be performed on the other side of the equation Step one: Isolate the unknown value.as well. Determine if multiplication orIf you “multiply by 2” on one side, you must “multiply division is needed.by 2” on the other side. Step two: Use division to divide bothIf you “divide by 3” on one side, you must “divide by sides by “8.”3” on the other side and so on. Step three: Simplify: x = 3
Find the value of an unknown using multiplication Do this example Solve the following: Find the value of “a” in the following equation. 2b = 40 a/3 = 6 1. Determine which operation is needed. Multiply both sides by 3 to isolate “a.” Division The left side becomes 1a or “a.” 2. Perform the same operation to both sides. Divide both sides by “2.” The right side becomes the product of 3. Isolate the variable and solve the equation 6 x 3 or “18.” b = 40/2 = 20 a = 18 Solve an Equation with Addition or Subtraction Don’t forget! Adding or subtracting any number from one side must 4 + x = 10 be carried out on the other side as well.Step one: Isolate the unknown value. Subtract “the given amount” from both sides. Determine if addition or subtraction is needed. Would solving 4 + x = 16 require addition or subtraction of “4” from each side?Step two: Use subtraction to isolate “x.” SubtractionStep three: Simplify: x = 6 Solve Equations Using More Than Do this example One Operation Solve the following: Isolate the unknown value. b - 12 = 8 Add or subtract as necessary first. 1. Determine which operation is needed. Multiply or divide as necessary second. Addition Identify the solution: the number on the side 2. Perform the same operation to both sides. opposite the unknown. Add “12” to both sides. Check the solution by “plugging in” the number 3. Isolate the variable and solve the equation. using the original equation. b = 8 + 12 = 20
Order of Operations “Undo the operations” When two or more calculations are written To solve an equation, we undo the operations, so symbolically, it is agreed to perform the operations we work in reverse order. according to a specified order of operations. 1. Undo the addition or subtraction. Perform multiplication and division as they appear from left to right. 2. Undo multiplication or division. Perform addition and subtraction as they appear from left to right. 7x + 4 = 39Try this example Equations Containing Multiple Unknown Terms 7x + 4 = 39First, undo the addition by subtracting 4 from each side. In some equations, the unknown value may occur more than once. And that becomes 7x = 35 The simplest instance is when the unknown value occurs inNext, divide each side by 7. two addends. And that becomes x = 35/7 = 5 For example: 3a + 2a = 25Verify the result by “plugging 5 in” the place of “x.” Add the numbers in each addend (2+3). 7 (5) + 4 = 39 Multiply the sum by the unknown (5a = 25). 35 + 4 = 39 Solve for “a.” (a = 5) Solve Equations ContainingTry this example ParenthesesFind a if: a +4a – 5 = 30 1. Eliminate the parentheses:Combine the unknown value addends. a. Multiply the number just outside the parenthesesa + 4a = 5a 5a – 5 = 30 by each addend inside the parentheses.“Undo” the subtraction. 5a = 35 b. Show the resulting products as addition or subtraction as indicated.“Undo” the multiplication. a=7 2. Solve the resulting equation.Check by replacing “a” with “7.It is correct.
Look at this example Tip! Solve the equation: Remove the parentheses first. 6(A + 2) = 24 5 (x - 2) = 45 Multiply “6” by each addend. Do me first ! 6 multiplied by A + 6 multiplied by 2 Show the resulting products. 6A + 12 = 24 5x -10 = 45 Solve the equation. 6A = 12 5x = 55 A=2 x = 11 Solve Equations That Cross products are Proportions An important property of proportions is that the cross A proportion is based on two pairs of related products are equal. quantities. A cross product is the product of the numerator of one The most common way to write proportions is to use fraction times the denominator of another fraction. fraction notation. A number written in fraction notation is also called a Example: 4/6 = 6/9 ratio. Multiply 4 x 9 = 6 x 6 36 = 36 When two ratios are equal, they form a proportion. Verify that two fractions Using Equations to form a proportion Solve Problems Do 4/12 and 6/18 form a proportion? There is a list of key words and what operations they imply in your textbook. Please refer to it.1. Multiply the numerator from the first fraction by the denominator of the second fraction. These words help you interpret the information and 4 x 18 = 72 begin to set up the equation to solve the problem.2. Multiply the denominator of the first fraction by the numerator of the second fraction. Example: “of” often implies multiplication. 6 x 12 = 72 “¼ of her salary” means “multiply her salary3. Are they equal? by ¼” Yes, they form a proportion.
Five-step problem solving approach for equations Use the solution planWhat you know. Known or given facts. Full time employees work more hours than part-time employees. If the difference is four per day, andWhat you are looking for? part-time employees work six hours per day, how Unknown or missing amounts. many hours per day do full-timers work?Solution Plan What are we looking for? Equation or relationship among known / unknown facts. Number of hours that FT workSolution What do we know? Solve the equation. PT work 6 hours;Conclusion The difference between FT and PT is 4 hours. Solution interpreted within context of problem. Use the solution plan Try this exampleWe also know that “difference” implies subtraction. Jill has three times as many trading cards as Matt. IfSet up a solution plan. the total number that both have is 200, how many cards does Jill have?FT – PT = 4FT = N [unknown] PT = 6 hours Use the five-step solution plan to solve this problem: 1. What are you looking for?N–6=4 2. What do you know?Solution plan: N = 4 + 6 = 10 3. Set up a solution plan. 4. Solve it.Conclusion: Full time employees work 10 hours. 5. Draw the conclusion. Solving a word problem with a Solution Plan total of two types of itemsWhat are you looking for? The number of cards that Jill has.What do you know? Diane’s Card Shop spent a total of $950 ordering 600 The relationship in the number of cards is 3:1; total is 200. cards from Wit’s End Co., whose humorous cards costSolution plan $1.75 each and whose nature cards cost $1.50 each. x (Matt’s) + 3x (Jill’s) = 200 How many of each style of card did the card shopSolve order? x + 3x = 200 4x = 200; x = 50 Use the solution plan to solve this problem.Conclusion Jill has “3x” or 150 cards.
What are you looking for? Organize the informationHow many humorous cards were ordered and how A total of $950 was What do youmany nature cards were ordered. know? spent.The total of H + N = 600 Two types of cards wereAnother way to look at this is: ordered.N = 600 – H The total number of cardsIf we let “H” represent the humorous cards, Nature ordered was 600.cards will be 600- H. The humorous cards costThis will simplify the solution process by using only one $1.75 each/nature cardsunknown: “H.” cost $1.50 each.Solution plan Solve the equation $1.75H + $1.50(600-H) = $950.00 Set up the equation by multiplying the unit price of each by the volume, represented by the unknowns $1.75H + $900.00 - $1.50H = $950.00 equaling the total amount spent. $0.25H + $900.00 = $950.00 $1.75(H) + $1.50 (600 – H) = $950.00 $0.25H = $50.00 Unit prices Volume Total spent H = 200 “unknowns” Conclusion Try this problemH = 200 Denise ordered 75 dinners for the awards banquet. Fish dinners cost $11.75 and chickenThe number of humorous cards ordered is 200. dinners cost $9.25 each. If she spent a total of $756.25, how many of each type of dinner didSince nature cards are 600 – H, we can conclude that she order?400 nature cards were ordered.Using “200” and “400” in the original equation Use the solution plan to organize the informationproves that the volume amounts are correct. and solve the problem.
Denise’s order Proportions The relationship between two factors is often$11.75(F) + $9.25(75-F) = $756.25 described in proportions. You can use proportions to solve for unknowns.$11.75 F + $693.75 - $9.25F = $756.25$2.50F + $693.75 = $756.25 Example: The label on a container of weed killer gives directions to mix three ounces of weed killer$2.50F = $62.50 with every two gallons of water. For five gallons of water, how many ounces of weed killer should youF = 25 use?Conclusion: 25 fish dinners and 50 chicken dinners wereordered. Use the solution plan ProportionsWhat are you looking for? Your car gets 23 miles to the gallon. How far can you The number of ounces of weed killer needed for 5 gallons go on 16 gallons of gas? of water.What do you know? 1 gallon/23 miles = 16 gallons/ x miles For every 2 gallons of water, you need 3 oz. of weed killer. Cross multiply: 1x = 368 milesSet up solution plan. 2/3 = 5/x Conclusion: You can travel 368 miles on 16 gallons ofSolve the equation. gas. Cross multiply. 2x = 15; x = 7.5Conclude You need 7.5 ounces of weed killer for 5 gallons of water. Direct Proportions FormulasMany business-related problems that involve pairs ofnumbers that are proportional involve direct Evaluate a formula.proportions. Find a variation of a formula by rearranging theAn increase (or decrease) in one amount causes an formula.increase (or decrease) in the number that pairs with it.In the previous example, an increase in the amount ofgas would directly and proportionately increase themileage yielded.
How to evaluate a formula Try this problem Write the formula. A plasma TV that costs $2,145 is marked up Rewrite the formula substituting known values $854. What is the selling price of the TV? for the letters of the formula. Use the formula S = C + M where S is the selling price, C is the cost, and M is Markup. Solve the equation for the unknown letter or perform the indicated operations, applying the S = $2,145 + $854 order of operations. S or Selling Price = $2,999 Interpret the solution within the context of the formula. Find a variation of a formula by rearranging the formula Try this problem Determine which variable of the formula is to be isolated The formula for Square Footage = Length x (solved for). Width or S = L x W. Solve the formula for W or Highlight or mentally locate all instances of the variable to width. be isolated. Isolate W by dividing both sides by L Treat all other variables of the formula as you would treat numbers in an equation, and perform normal steps for The new formula is then: S/L = W solving an equation. If the isolated variable is on the right side of the equation, interchange the sides so that it appears on the left side.Simultaneous Equations Simultaneous EquationsExercise 1: Exercise 2:Solve for x & y: x 2y 6 Solve for x & y: 2 x 3 y 12 x 3y 4 5x 4 y 23
Simultaneous Equations Simultaneous EquationsExercise 3: Exercise 4:A cup and a saucer cost $5.25 together. A cup and A shop sells bread rolls. If five brown rolls and six two saucers cost $7.50. Find the cost of a cup. white rolls cost $2.94 and three brown rolls and four white rolls cost $1.86 find the cost of each type of roll.