PDF. 3x+7y =27 3x+21=27 3x =6 x =2 Asbefore . 8"iR1Elx~af8ux;j#8>6~P~J Jj;MZG2Ez(SayfUDt7$A6n5~'#f) 3hWXrC}|SYA*}9&ZOBO}*Xz=>yuxumG^op_0PW6q_k^. /zCJHOJJa<= Linear Equations a. Determine whether a number is a solution to an equation. If the two lines intersect at a single point, then there is one solution for the system: the point of intersection. VERIFYING SOLUTIONS A linear equation is made up of two expressions that are equal to each other. Linear Equations Notes MODULE - 1 Algebra Mathematics Secondary Course 139 5 LINEAR EQUATIONS You have learnt about basic concept of a variable and a constant. Be sure to look at the notes on each slide for additional instruction and answers. 0000002745 00000 n notes for solving linear equations - Free download as PDF File (.pdf), Text File (.txt) or read online for free. trailer Consider non-autonomous equations, assum-ing a time-varying term bt.2 In general, the solutions of these equations will take the functional form of bt. 252 0 obj <>stream Solve the following equations for x: 1) x + 5 = 12 2) x - 11 = 19 3) 22 - x = 17 4) 5x = -30 5) = 3 6) x = - 8 1184 0 obj <>/Filter/FlateDecode/ID[<9D126D529814A74B9CAAD449527FF8DB><0A0F665A7A87EB43B68CD8BCE126640B>]/Index[1161 75]/Info 1160 0 R/Length 113/Prev 319934/Root 1162 0 R/Size 1236/Type/XRef/W[1 3 1]>>stream Detailed steps are provided along with many examples and word problems. Determine whether the resulting equation is true. G]NvxmVH-m6 QQ}`Dl@~VJ. A linear equation in one variable is an equation with the exponent 1 on the variable. Solving equations worksheets cazoom maths 50 linear worksheet pdf chessmuseum template library algebra in one variable free tutorial for beginners with answer key edboost fresh word problems math methods homework packet solutions of two variables tassee systems graphing method notes grade 7 otosection Solving Equations Worksheets Cazoom Maths Solving Equations Worksheets Cazoom Maths 50 . 0000028048 00000 n hb```b``fd`e``gd@ A+@FE-cNfy1H1008)&>(`"b@J|5~nhqJ^yKtkRw~gaG$}!Zn[_$7|WS|_//WB^n}/{jEW&^+sEpUFrm3 0h` ;B"`1AaXX^f0 4iNeeD~_`/] @@ ~OJ%C`m V WAp;0P7>`d>]bzu~3MDV#N7=lH[Yo(Cu{jl Decide on what kind of eSignature to create. The opposite of addition is subtraction and the opposite of multiplication is division. The resulting equation should have only one variable, not both x and y. Solving Linear Equations in One Variable Guided Notes 8.EE-7. coefficient: Sometimes a variable is multiplied by a number. 11 5z = 6 (5z + 4) 6. II. a) Subtract 6 from 6 + m, and subtract 6 from 15 . The least common denominator is _____ Step 2: Multiply every term in the equation by the least common denominator. 4 0 obj When a nonlinear system consists of a linear equation and a quadratic equation, the graphs can intersect in zero, one, or two points. Case 2: Parallel Lines oT@zq`gwp#%p_+|xW8 (C;_w' 2B)Zyx}x QdOW+JTM*oyI)141nr. The opposite of addition is subtraction and the opposite of multiplication is division. k(3YuN`Q>zWN >"KDys$S'dj&oN_ov$AaIC^$W8zBrc dCM@ngT&6OIGrSzR5X? 0000104184 00000 n Linear equation: 2= 3+ 1 x a _ =b Solve ax= b Check a x _ =bx a _b Solve Check x b =c =c a(x+ b) =c Solve ax= b + cx ax b= cx + d a(bx c) (ex f) Check Key Words Solving Single Step Equations: To solve single step equations, you do the opposite of whatever the operation is. is a matrix with two rows and three columns. <> 0000103044 00000 n 15x +35y = 135 15x +6y =48 29y =87 fromwhich y = 87 29 =3 IfwesubstitutethisresultinEquation(1)wecanndx. o9nUs$Jl_Q0%` XlPV-_\ahKqJ/B ZZMV~AhV8t w`@0byy=m\:bpMXH'B~!F/bqMf0#H]OK#Y`N)${QfA@j}Tg' "w7q.nJ~!n0u8e DA(m 8Lo8;rf?(|5 :M|Z} endstream endobj 216 0 obj <> endobj 217 0 obj <> endobj 218 0 obj <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>> endobj 219 0 obj <> endobj 220 0 obj <> endobj 221 0 obj <> endobj 222 0 obj [/ICCBased 243 0 R] endobj 223 0 obj <> endobj 224 0 obj <> endobj 225 0 obj <> endobj 226 0 obj <>stream e. Solve for the other variable. << /Length 5 0 R /Filter /FlateDecode >> SOLVING SYSTEMS OF LINEAR EQUATIONS SOLVING SYSTEMS OF LINEAR EQUATIONS An equation is said to be linear if every variable has degree equal to one (or zero) is a linear equation is NOT a linear equation Review these familiar techniques for solving 2 equations in 2 variables. "LqmYkl`D5 V'X`5jD+UV.+T+X`S+Ph:.Xh3_`y`Q/,X`9e,8ae . Solving Linear Equations: Variable on Both Sides Solve each equation. l]H3m\D,!U Step 2: Identify the solution. <>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/Annots[ 22 0 R] /MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> Step 2 Add or subtract the equations to eliminate one of the variables. $2.00. Simplify the expressions on both sides of the equation. But first we need to define some important terms: variables: Variables are symbols that stand for an unknown quantity, they are often represented with letters, like x, y, or z. Open navigation menu Close suggestionsSearchSearch enChange Language close menu Language English(selected) espaol portugus Deutsch franais italiano Romn 1161 0 obj <> endobj Choose My Signature. . Step 4: Solve for the variable using the steps to solve an equation @H &'!83C @@@fKWU]lZSU}]vw=1CD%k qk/x+!ooqHY/ SG i:&nBHjM[\" h\A[.|?y E>swOg>)BM#S18HWRC4?BtUU_1~j,wm4"iD}}?29[8lTg,V~+n/9I7(&% uCk@ZrZxV: -~)(5hgdTZ$RC\!\&A H_K%D}\,^ H[`$*IEl`$X/: JF %X-* $X) stream %a tRbc! BPP. Adding the same thing to both sides of an equation. 10) How could you solve for the variable m in this equation? Subtracting the same thing from both sides of an equation. 0000028449 00000 n 11) What method would you use to solve 6 + m = 15? View Chapter 2 Blank Class Notes.pdf from MATH 250BC at Cypress College. Step 1. *K?j3v2}8BNht^H?1:O'wt?x&f@>z*A`wFt#qs:txt}AZw?x)C+ -YO\ww.o?Jw~^)=ILG7 _>eY ^?)Odo~~Y#uaP27#IDFz\}Hk.Q+$BtPA7E=}P}%^\/< \'U\s.m5 ch:{ /%/Yc(?q_#^~_@NO |K-?|K>ykxN. endstream endobj 1162 0 obj <. 2.1.2. endstream endobj startxref Solving Linear Equations. these notes, it is preferable to view solving Equation (3) as a question about the linear map T L(Fn,Fm) having so-called canonical matrix A, as explained in the next section. Example: Step 1: Find the least common denominator. xio9{ab|(D3b3N]`h "ICe~gWuWuFw]ggo'|*M0(D Multiplying both sides of an equation by the same non-zero thing. Solving a System of Linear Equations by Elimination Step 1 Multiply, if necessary, one or both equations by a constant so at least one pair of like terms has the same or opposite coefficients. The unknowns, which we will seek (for now) among the real numbers, are denoted by x 1;:::;x n. An equation in these nunknowns is called linear, if it is of the form a 1x 1 + :::+ a nx n= b where a 1;:::;a Scribd is the world's largest social reading and publishing site. These are all linear equations: y = 2x + 1 : 5x = 6 + 3y : y/2 = 3 x: Let us look more closely at one example: Example: y = 2x + 1 is a linear equation: The graph of y = 2x+1 is a straight line . @l`s u:=Dj`#9H Classwork Page 5 Homework Page 6 Extra Practice Page 7-10 Worked example 1: Solving linear equations Solve for x: 4 ( 2 x 9) 4 x = 4 6 x Expand the brackets and simplify 4 ( 2 x 9) 4 x = 4 6 x 8 x 36 4 x = 4 6 x 8 x 4 x + 6 x = 4 + 36 10 x = 40 Divide both sides by 10 that graph as straight lines) are simpler than non linear equations , and the simplest linear system is one with two equations and two variables. There are three variants; a typed, drawn or uploaded signature. {grUCJLj/VAe^x"O55N~5z0]P2 8'[1\>X_YT? 2m 5 12 3 1. w w1 77 2. x 2x 5 63 3. x 5 82 Standard(s): MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with . Solve for x: 1) x + 5 = 12 2) x - 11 = 19 3) 22 - x = 17 4) 5x = -30 5) (x/-5) = 3 6) x = - 8 Solving Multi-Step Equations: 3. If bt is an exponential or it is a polynomial of order p, then the solution will, %PDF-1.5 The methods for solving systems of linear equations can also be used to solve systems of nonlinear equations. 0000075129 00000 n No variable in a linear equation can have a power greater than 1. Linear equations generally have only one solution for each variable. 0000030774 00000 n In broad terms, vectors are things you can add and linear functions are functions of vectors that respect vector addition. We do this by adding or subtracting to each side of the equation but always keep it balanced! K+A$X*AK$X," $X( K0OS fK0Kv $hE %AML`S%hAz $Fj $Br $(Db LHB %( _yD%H W %I K~ |x%H%Sv lX%H`$Qz th%HJH@% @%KJCK7 *_$x_$] -[^w!o%xSHKU ^W,K(R%xNg%HpZK3 ~*O$?@S|_I] NJ) -JpBG$`XI K|]IU +/}+%G%K;%_ &%Jpf >#MIp7HpIpHpWIp$B%L`Jp$X`$) KMJp[$,& 6JA`k$_O`HLtKD,`$/\ H)A%%AmJry5#Ma#^DF]>kAe"uP(rHN>.lGvKd An equation must always be balanced, whatever you do to the left-hand side, you must also do to the right-hand side. (Ordered pair where the lines intersect) Step 1: Equation a: y = 3x + 1 Use half a sheet of 8.5 11 paper to create a pocket for storing Key Words and examples of linear equations. 3 0 obj hbbd```b``kdd:"9*@$s fIo-"g`T@dd|L,`qQ Make a table and find points to plot. No b. Steps We May Take to Solve Equations Every step taken to solve an equation or a system of equations may be categorized as one of the following. Here is a general strategy to use when solving linear equations. Divide by the coefficient using the multiplication property of equations. 8.1 - Solve Equations Using the Subtraction and Addition Properties of Equality. . Now suppose we have m linear equations in n unknowns. Follow the step-by-step instructions below to eSign you're solving linear equations sudoku answer key: Select the document you want to sign and click Upload. All linear equations eventually can be written in the form ax + b = c, where a, b, and c are real numbers and a 0. #2 0000031725 00000 n 10 3 2 3x m 2. 5 (31) 2 = 20 stream _``>|H+X*? 3 0000102362 00000 n d. The solution to an equation is the set of all values that check in the . 0000002309 00000 n F[-*QOV Solving linear equations A linear equation is an equation that contains letters and numbers, for example \ (3x + 10 = 16\). Solving Systems Of Equations Word Problems. 2 0 obj In Example 4, you solve the equation given in Example 2, using the four preceding steps to find the solution to the equation. to non-autonomous equations and to systems of linear equations. %PDF-1.3 c. Solve the equation. Example 4: Solve the following equation: 4 ( x 3) + 6 = 14 + 2 x. Solving Linear Equations - Fractions Objective: Solve linear equations with rational coecients by multi-plying by the least common denominator to clear the fractions. This is often referred to as a "two by three matrix", a "23-matrix", or a matrix of dimension 23.Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be . %PDF-1.4 % of equations, such as Solving Systems of Linear Equations Substitutions. endobj (x, y) 3 2 26 2 1 When x increases, y increases twice as fast, so we need 2x; Unit 1A, Pre-Lesson, Solving Linear Equations, A-REI.2.3 Learning Target: Students will solve linear equations. 8Eaq|jBW[Tv?q,|IfM} 0000003144 00000 n \(4x - 7\left( {2 - x} \right) . E8S=;]NUR:RuzSQ{`Bw5U.n7Vokt ozi5-D"3I}}aT*SP98T:yy> LrfN?=^'/Vs8q4,f ';x I~E+T`~?. b. Find the x- and y- intercepts. Use the four steps to solving a linear equation, as follows: 0000050072 00000 n The General Form of a basic linear equation is: ax b c. c. To Solve: the goal is to write the equation in the form variable = constant. The same techniques will be extended to accommodate larger systems. 0000003535 00000 n startxref 1. Yes miL2872X_ch03_177-254 09:22:2006 02:15 PM Page 178 CONFIRMING PAGES 4 0 obj <> Monday (9/8): Notes: writing equations from word problems Thursday (9/4): 1.2 video- multi-step word problems Friday - Wednesday (8/29-9/3): 1.2 Notes - Solving multi-step linear equation Video of how to solve multi-step equations Wednesday (8/27): 1.1 Notes: Solving one-step linear equations This worksheet includes guided examples and independent practice with an answer bank attached. Linear equations are equations that have two variables and are a straight line when graphed, based on their slope and y-intercept. In the equation, 7 added to a number gives 15 Solving the equation means, finding the value of the variable that makes the . A system of nonlinear equations is a system in which at least one of the equations is nonlinear. Chapter 5 - Systems of LInear Equations Chapter 5 Calendar, Overview, and Vocab 5-1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING (Monday, January 8) Notes: 5-1_solving_by_graphing_notes.pdf Assignment: 5-1_solving_by_graphing_worksheet.pdf 5-2 SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION (Tuesday, January 9) Notes: 5-2_solving_by_sub_notes.pdf Here is a set of practice problems to accompany the Linear Equations section of the Solving Equations and Inequalities chapter of the notes for Paul Dawkins Algebra course at Lamar University. 0000001667 00000 n hb```f``Ib`g``e`d@ A qF| 6;e3>_*QkP_TN-Qr%TmH Idntip90dVN`q/H|;+mqKG! Teacher Note Be sure to classify each system as consistent or inconsistent and dependent or independent. 215 38 When solving equations with fractions, you want to find a way to eliminate the fraction. Solving One-Step Equations A. The 11th Commandment (for equations): To solve an equation means to find every number that makes the equation true. 0000103901 00000 n Paul's Online Notes. x]KRbXh #@vtq8k8qHCh$y2WGxC#B9^z;7;~|{?Tc[ioHa}?Fi]+#?"&5 1o+MnHR*:~}7K($'` V]L/{F2K {1k^KAS~A]sF]~l0 q#^;/Y0]WntB_}6}w?>MC?f?]D1E)e|-`|ohLM~?uoSlFOS;xLG&8B>9QcPuONqpL_|:! To solve single step equations, you do the opposite of whatever the operation is. Step 5 Staple the three booklets and the pocket you made into the Foldable from Step 1 as shown. One way to solve a system of linear equations is by graphing each linear equation on the same -plane. This will give you one of the coordinates. 0000001410 00000 n Step 5: Solve for the other coordinate. cEXyLL0mf|xIS]'&la8x71,% f. Write the solution as an ordered pair. stream Common Core State Standards - Grade 8 Mathematics11 pages of guided notes for the entire Solving Linear Equations in one variable unit. 215 0 obj <> endobj ez@G`h|_xc5Ve0*pvv54#q =S Type 1: one-step equations (the simplest possible, such as x + 6 = 19 or 6x = 17 or x/7 = 18) nonnegative solutions only Type 2: other one-step equations (such as 4 = 8 x) nonnegative solutions only Type 3: one-step equations where you first need to simplify an expression on one side (such as 4x = 19 7 or 10x 2x = 16) If it is true, the number is a solution. We can solve these problems as we have in the past. 0000001476 00000 n <> % Solving Linear Systems To solve a linear system analytically it is typically transformed into a system for which a solution can be easily computed Diagonal system Triangular systems b a 1,10 0 0 0a n,n x = b x 1= b 1 a 1,1 x n= b n a n,n a 1,1 a 1,n 0 0a n,n x = b x n= n a n,n x n1= b n1a n1,n x n a n1,n1 Solve each of the following equations and check your answer. <]/Prev 251948>> 1.1.3 Systems of linear equations. a$E: %%EOF You will also learn to solve linear equations in two variables using graphical as well as algebraic methods. Linear Equations. %PDF-1.3 Step 3: Reduce each term to create a "denominator free" equation. When this is done, one of three cases will arise: Case 1: Two Intersecting Lines . There are 3 possible scenarios for linear systems: Exactly One Solution Infinite Solutions No Solution y = x - 4 y = 2x - 2 y = 3x + 5 y = 3x + 5 y = 3x + 1 y = 3x + 5 To solve an equation graphically, all we do is graph both equations and find where they Often when solving linear equations we will need to work with an equation with fraction coecients. 0000000016 00000 n 0000102628 00000 n Prior Knowledge % The solution to a system of linear equation occurs where the two lines intersect. @/W' ,8bg&sz/S _r6@8G6V This is Step 3: Solve the new equation. 2 = (w 8) 3. Linear algebra is the study of vectors and linear functions. In other words, you are trying to __isolate__ the variable. *|w;ZM~+mwwy1! SECTION 3.1: LINEAR EQUATIONS A. These are also known as firstdegree equations, because the highest exponent on the variable is 1. 3 x 1 8 2 4. by. B,. xref Solutions to Systems of Linear Equations 2. %%EOF 8 = 8c 4 (c + 8) 5. Section 3.1 Solving Systems of Linear Equations by Graphing Concepts 1. % <>>> 1 0 obj endobj ;.6Qd`]4[,&6 Non-autonomous equations, lags and leads. 37 = 3 + 5 (k + 6) 2. Dependent and Inconsistent Systems of Linear Equations 3. 1.2 The Canonical Matrix of a Linear Map Let m,n Z+ be positive integers. Share with the class that today . 0000028708 00000 n Solve for x: 1) x + 5 = 12 2) x - 11 = 19 3) 22 - x = 17 4) 5x = -30 5) (x/-5) = 3 6) x = - 8 Solving Multi-Step Equations: Solving Linear . Key included! This number is called the . In general, in order to solve an equation, you want to get the variable by itself by undoing any operations that are being applied to it. Teach as directed in the PowerPoint. {Quick steps: Solve, Substitute, Solve, Substitute, Solve, Write the solution} Ex 1: and The solution is . 0000047105 00000 n 21UYFE?KELj1,c[_r55 ~nq1 1) 6 r + 7 = 13 + 7r 2) 13 4x = 1 x 3) 7x 3x + 2 = 8x 8 4) 8 x = x 4x 5) 14 + 6b + 7 2b = 1 + 5b 6) n + 2 = 14 n 7) n 3n = 14 4n 8) 7a 3 = 3 + 6a 9) 5 + 2x = 2x + 6 10) 10 + x + 4 5 = 7x 5 . 0000025895 00000 n e@ JXLJPs!2B1 0000047331 00000 n 0000048872 00000 n By the end of the unit you will also be able to work with linear systems with three equations and three variables. 0000047898 00000 n 0000051217 00000 n 0000055170 00000 n Solving equations with variables on both sides. It is time to de ne the term linear equation precisely. 0 Substitute the number for the variable in the equation. Equations. 0000032093 00000 n 0000025674 00000 n d. Substitute the value you just found into the first equation. Substitute the result into the other equation to replace one of the variables. Ppt Chapter 1 Systems Of Linear Equations Powerpoint Presentation Free Id 1977234. 1235 0 obj <>stream 0000001056 00000 n B Then, given a choice of bases for the vector spaces Fn OBJECTIVES After studying this lesson, you will be . The goal of this text is to teach you to organize information about vector spaces in a way that makes problems involving linear functions of many variables easy. %PDF-1.5 % There are step by step directions with fill-in the blanks to help students who may not write as fast at the rest of their peers. Important Rules for Solving Equations Rule #1) When you solve an equation, your goal is to get the __variable__ alone by itself on _one_ __side_ of the equation. I. 42 = 8x + 13x 4. Linear equations examples (one variable): Solve 5x - 3/2 = 7 Solution: 5x - 3/2 = 7 5x = 7 + 3/2 5x = 17/2 x = 17/10 Solving linear Equations in Two Variables: @/OdzL`.C0 f"3 fd-kG4"S)*6"` HK endobj 0 5 0 obj Step 3 Solve the resulting equation. 2 Linear Equations 6 3 Matrix Algebra 8 4 Determinants 11 5 Eigenvalues and Eigenvectors 13 6 Linear Transformations 16 7 Dimension 17 8 Similarity and Diagonalizability 18 9 Complex Numbers 23 10 Projection Theorem 28 11 Gram-Schmidt Orthonormalization 29 12 QR Factorization 31 13 Least Squares Approximation 32 14 Orthogonal (Unitary . Instructional Activities Step 1 - Discuss the methods they have learned for solving systems of equations (graphing and substitution). 0000001899 00000 n It does not contain any \ (x^2\) or \ (x^3\) terms. 0000090734 00000 n 8th Grade Math Guru. Benefits of Linear Equations Worksheets ability to solve systems of equations by graphing. Solving Multi-Step Equations CC Standards 7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any . example: solve forx: x 4 = -3 x 4 + = -3 + subtraction property of equality for all real numbersa,b, andc, ifa=b, then a c=b c. example: solve forx: x+ 5 = -1 x+ 5 = -1the value (s) that satisfy an equation solve one-step and two-step linear equations in one variable. This product contains guided notes for solving equations with variables on both sides of the equal sign. Solving Systems of Linear Equations by Graphing Skill Practice Answers 1a. x\i_~/iq (gA.qHNMfR$Kg]-,x"^ OA>_~_J_>}}M' /[xP8q+N~??o0FhwQ0pQOXO!"W/ _ta+qn(U 3}? C( s@`JJ x t\6ZuNnI%Y-lf ! 0000003232 00000 n Pdf 6 iteration solution of linear equations notes packet diffeial solving systems with algebra 1 math 221 lecture math221 5 numerical methods ceng 2002 and solved m192hwk5 192 homework by Basic Principles of Solving Linear Equations When you are solving a linear equation like 2 ( x + 3) 4 =14 x, these are the basic principles: You are done when the variable is alone on one. Step 2. To solve single step equations, you do the opposite of whatever the operation is. variable solution inverse balanced properties of equality adding the same 0 = 5y 2y 7. +2 1 = 3 8. 6 4 3. 12 4 m a) Multiply both sides of the equation by 4. b) Divide both sides of the equation by 4. c) Multiply both sides of the equation by 12. d) Divide both sides of the equation by 12. 2.1 Systems of Linear Equations Example: Solve: x + 3y = 1 2x - 5y = -9 Graphically: Equations in 2 variables represent lines 1 1.1 Solving One-Step Linear Equations In this section, we are going to learn how to solve one-step linear equations. The opposite of addition is subtraction and the opposite of multiplication is division. 0000003282 00000 n Independent Practice Solve for the variable in the equations below. 0000052462 00000 n A linear equation is an equation for a straight line. Put equations into slope intercept form: * Add or Subtract the x-term * Divide all terms by # in front of y Graph using y-intercept (b) and slope (m). 4. -H. 0000030000 00000 n Clear fractions or decimals. Step 6: Write the solution as an ordered pair. Step 4: Substitute the result from Step 3 into either of the original equations. For solving the linear equations in one variable, inverse Mathematical operations are used. A linear equation may have one or two variables in it, where each variable is raised to the power of 1. Linear equations may look like x + 6 = 4, or like 2 a - 3 = 7. 0000026637 00000 n >X ,HE{O\HI3Q\{WwD{[(_";O~+o{Cd.kZWEWDWE^yYQ=#2Os~&OEc#~(;%Ew;)w{Jd~['{B%Oq vT##f`iQ%#>h=$=(`_"'We.KE m9qVn,gDDjHAn;gvn@*v#uh^1\Dx%b To eliminate the fraction, multiply by a Common Denominator 1. It is assumed that you are familiar with . 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