Linear Equations - Word Problems - CCfaculty.org

1 Equations - Word ProblemsWord Problems can be tricky. Often it takes a bit of practice to convert theenglish sentence into a mathematical sentence. This is whatwe will focus on herewith some basic number Problems , geometry Problems , and parts few important phrases are described below that can give us clues for how to setup a problem . A number(or unknown, a value, etc) often becomes our variable Is(or other forms of is: was, will be, are, etc) often represents equals (=)xis5becomesx= 5 More thanoften represents addition and is usually built backwards,writing the second part plus the firstThree more than a number becomesx+ 3 Less thanoften represents subtraction and is usually built backwards aswell, writing the second part minus the firstFour less than a number becomesx 4 Using these key phrases we can take a number problem and set upand equationand 28 less than five times a certain number is 232.

2 What is the number?5x 28 Subtraction is built backwards,multiply the unknown by55x 28=232 Is translates to equals+28+28 Add 28 to both sides5x=260 The variable is multiplied by555 Divide both sides by5x=52 The number is same idea can be extended to a more invovled problem as shown in more than three times a number is the same as ten less than six times thenumber. What is the number3x+15 First,addition is built backwards6x 10 Then,subtraction is also built backwards3x+15= 6x 10 Is between the parts tells us they must be equal 3x 3xSubtract3xso variable is all on one side15= 3x 10 Now we haveatwo step equation+10+10 Add 10 to both sides25= 3xThe variable is multiplied by333 Divide both sides by3253=xOur number is253 Another type of number problem involves consecutive numbers that come one after the other, such as 3, 4, 5.

3 If wearelooking for several consecutive numbers it is important to first identify what theylook like with variables before we set up the equation. This is shown in sum of three consecutive integers is 93. What are the integers?FirstxMake the first numberxSecondx+ 1To get the next number we go up one or+ 1 Thirdx+ 2 Add another1(2total)to get the thirdF+S+T=93 First(F)plus Second(S)plus Third(T)equals 93(x) + (x+ 1) + (x+ 2) =93 ReplaceFwithx, Swithx+ 1,andTwithx+ 2x+x+ 1 +x+ 2 =93 Here the parenthesis aren + 3 =93 Combine like termsx+x+xand2 + 1 3 3 Add3to both sides3x=90 The variable is multiplied by333 Divide both sides by3x=30 Our solution forxFirst 30 Replacexin our origional list with 30 Second(30) + 1 =31 The numbers are 30,31,and 32 Third(30) + 2 =32 Sometimes we will work consective even or odd integers, rather than just consecu-tive integers.

4 When we had consecutive integers, we only hadto add 1 to get tothe next number so we hadx,x+ 1, andx+ 2for our first, second, and thirdnumber respectivly. With even or odd numbers they are spacedapart by two. Soif we want three consecutive even numbers, if the first isx, the next numberwould bex+ 2, then finally add two more to get the third,x+ 4. The same is2true for consecutive odd numbers, if the first isx, the next will bex+ 2, and thethird would bex+ is important to note that we are still adding 2 and 4 evenwhen the numbers are odd. This is because the phrase odd is refering to ourx,not to what is added to the numbers. Consider the next two sum of three consecutive even numbers is 246. What are thenumbers?FirstxMake the firstxSecondx+ 2 Even numbers,so we add2to get the nextThirdx+ 4 Add2more(4total)to get the thirdF+S+T=246 SummeansaddFirst(F)plusSecond(S)plusThir d(T)(x) + (x+ 2) + (x+ 4) =246 Replace eachF , S ,andTwith what we labeled themx+x+ 2 +x+ 4 =246 Here the parenthesis are not needed3x+ 6 =246 Combine like termsx+x+xand2 + 4 6 6 Subtract6from both sides3x=240 The variable is multiplied by333 Divide both sides by3x=80 Our solution forxFirst 80 Replacexin the origional list with (80) + 2 =82 The numbers are 80,82,and (80) + 4 =84 Example three consecutive odd integers so that the sum of twice the first, the secondand three times the third is the firstxSecondx+ 2 Odd numbers so we add2(same as even!)

5 Thirdx+ 4 Add2more(4total)to get the third2F+S+ 3T=152 Twicethefirstgives2 Fandthreetimesthethirdgives3T2(x) + (x+ 2) + 3(x+ 4) =152 ReplaceF , S ,andTwith what we labled them2x+x+ 2 + 3x+12=152 Distirbute through parenthesis6x+14=152 Combine like terms2x+x+ 3xand2 +14 14 14 Subtract 14 from both sides6x=138 Variable is multiplied by666 Divide both sides by6x=23 Our solution forxFirst 23 Replacexwith 23 in the original listSecond(23) + 2 =25 The numbers are 23,25,and 27 Third(23) + 4 =273 When we started with our first, second, and third numbers for both even and oddwe hadx,x+ 2, andx+ 4. The numbers added do not change with odd or even,it is our answer forxthat will be odd or example of translating english sentences to mathematical sentencescomes from geometry.

6 A well known property of triangles is that all three angleswill always add to 180. For example, the first angle may be 50 degrees, the second30 degrees, and the third 100 degrees. If you add these together, 50+30+100=180. We can use this property to find angles of second angle of a triangle is double the first. The third angle is 40 less thanthe first. Find the three nothing given about the first we make thatxSecond2xThe second is double the first,Thirdx 40 The third is 40 less than the firstF+S+T=180 All three angles add to 180(x) + (2x) + (x 40) =180 ReplaceF , S ,andTwith the labeled + 2x+x 40=180 Here the parenthesis are not 40=180 Combine like terms, x+ 2x+x+40+40 Add 40 to both sides4x=220 The variable is multiplied by444 Divide both sides by4x=55 Our solution forxFirst 55 Replacexwith 55 in the original list of anglesSecond2(55) =110 Our angles are 55,110,and 15 Third(55) 40=15 Another geometry problem involves perimeter or the distance around an example, consider a rectangle has a length of 8 and a widthof 3.

7 Their aretwo lengths and two widths in a rectangle (opposite sides) sowe add8 + 8 + 3 +3 =22. As there are two lengths and two widths in a rectangle an alternative tofind the perimeter of a rectangle is to use the formulaP= 2L+ 2W. So for therectangle of length 8 and width 3 the formula would give,P= 2(8) + 2(3) =16+6 =22. With Problems that we will consider here the formulaP= 2L+ 2 Wwill perimeter of a rectangle is 44. The length is 5 less than double the the will make the lengthxWidth2x 5 Width is five less than two times the lengthP= 2L+ 2 WThe formula for perimeter ofarectangle(44) = 2(x) + 2(2x 5)ReplaceP , L,andWwith labeled values44= 2x+ 4x 10 Distribute through parenthesis44= 6x 10 Combine like terms2x+ 4x+10+10 Add 10 to both sides54= 6xThe variable is multiplied by666 Divide both sides by69 =xOur solution forxLength9 Replacexwith9in the origional list of sidesWidth2(9) 5 =13 The dimensions of the rectangle are9by have seen that it is imortant to start by clearly labeling the variables in ashort list before we begin to solve the problem .

8 This is important in all wordproblems involving variables, not just consective numbersor geometry is shown in the following sofa and a love seat together costsS444. The sofa costs double the love much do they each cost?Love SeatxWith no information about the love seat,this is ourxSofa2xSofa is double the love seat,so we multiply by2S+L=444 Together they cost 444,so we add.(x) + (2x) =444 ReplaceSandLwith labled values3x=444 Parenthesis are not needed,combine like termsx+ 2x33 Divide both sides by3x=148 Our solution forxLove Seat 148 Replacexwith 148 in the origional listSofa2(148) =296 The love seat costsS148 and the sofa careful on Problems such as these. Many students see the phrase double andbelieve that means we only have to divide the 444 by 2 and getS222 for one orboth of the prices.

9 As you can see this will not work. By clearly labeling the vari-ables in the original list we know exactly how to set up and solve these and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( ) - Word When five is added to three more than a certain number, the result is is the number?2. If five is subtracted from three times a ce3rtain number, the result is 10. Whatis the number?3. When 18 is subtracted from six times a certain number, the result is is the number?4. A certain number added twice to itself equals 96. What is the number?5. A number plus itself, plus twice itself, plus 4 times itself, is equal to is the number?6. Sixty more than nine times a number is the same as two less than ten timesthe number.

10 What is the number?7. Eleven less tahn seven times a number is five more than six times the the Fourteen less than eight times a number is three more than four times thenumber. What is the number?9. The sum of three consectutive integers is 108. What are theintegers?10. The sum of three consecutive integers is 126. What are the integers?11. Find three consecutive integers such that the sum of the first, twice thesecond, and three times the third is The sum of two consectutive even integers is 106. What arethe integers?13. The sum of three consecutive odd integers is 189. What arethe integers?614. The sum of three consecutive odd integers is 255. What arethe integers?15. Find three consecutive odd integers such that the sum of the first, two timesthe second, and three times the third is The second angle of a triangle is the same size as the first angle.