12.1 Systems of Linear equations: Substitution and Elimination
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1 . Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables is a pair of numbers that satisfies all the equations in the sstem. Eample: is a sstem whose solution is the pair ) There are two tpes of sstems: consistent have a solution) and inconsistent do not have a solution) Consistent sstem can be dependent have infinitel man solutions) and independent have onl one solution) Sstem of two linear equations with two variables can be represented b two straight lines. In the sstem above one line is represented b the equation + = and the other line b the equation + = If the lines intersect the point of intersection is the solution of the sstem. The sstem is consistent independent. If the lines are identical then there are infinitel man solutions. Each point on the line is a solution. The sstem is consistent dependent. If the lines are parallel there is no solution. The sstem is inconsistent. Sstem of equations can be solved graphicall ou need a ver accurate graph for that) b substitution b elimination addition) using matrices we will not stud this method) and using Cramer s Rule we ll learn it in sec.). Graphical method: Carefull graph both equations in the same coordinate plane. Use graph paper and a ruler. The intersection of the two graphs if an is the solution. Eample: Solve b graphing Line + = = - + slope m = - intercept : 0) Line + = = - + slope m = - -intercept : 0)
2 Substitution method: Solve one of the equations for one of the variables sa substitute to the other equation and solve for. Then go back to the first equation to find value of. Eample: Solve ) substitute ) 0 solution : 0) Elimination addition) method: Multipl one or both equations b suitable numbers so that after the multiplication is done coefficients of one of the variables sa are opposite numbers. Add both equations side b side to eliminate that variable. Solve the resulting equation for. Substitute in one of the equations and solve for. Eample: we eliminate ) Multipl first equation b so the coefficients of are and - Add side b side Solve for Substitute = into the first equation Solve for Solution: 0) Remark: - if after substitution or elimination ou obtain an equation that is alwas true like 0 = 0) then the sstem has infinitel man solutions. The line represented b either equation contains all solutions - If after substitution or elimination ou obtain an equation that is alwas false like 0 = ) then the sstem has no solutions.
3 Sstems of three linear equations with three variables Eample: Solution of such a sstem is a triple of numbers abc) that when substituted for respectivel in each equation makes each equation a true statement. A sstem of three linear equation can have one solution infinitel man solutions and no solutions Sstem of three equations with three variables can be solved b substitution and elimination. Substitution method Eample: Solve ) Solve one of the equations for one of the variables choose an equation that is easiest to solve for one of the variables). First equation can be easil solved for : = - + ) Substitute in the remaining two equations with the epression obtained in step ) ) ) ) Simplif the two equations. Notice that the last two equations contain onl and. ) Solve the resulting sstem of two equations with two variables and ) The solution is = = - ) use the first equation to find value of : = + = - = ) The solution is -)
4 Elimination method Eample: Solve the sstem The idea is to eliminate one of the variables b multipling the equations b suitable constants adding two equations side b side and reducing the sstem to a sstem with two equations and two variables. We ll eliminate. ) Multipl the first equation b -) and add to the second equation = - + = - + = -7 this is now the equation ) ) Multipl the first equation b -) and add to the third equation = 0 + = -7 = this is now the equation ) ) Solve the sstem 7 7 Using elimination we get = = - ) Use the first equation to find - - = - = ) solution is -) Eample: Infinitel man solutions) Solve Using the elimination method eliminate using the first equation) we get the sstem Solving b elimination the sstem leads to the equation 0 = 0. This means that there are infinitel man solutions. To find the solution set notice that the sstem reduces to or. We get = + and = - = +) - - = -. Therefore if is an real number then = - = + = is a solution. The solution set is {) = - = + is an real number}
5 . Sstem of linear equations: determinants and Cramer s Rule Consider a sstem of equations a b s c d t Let s use the Elimination method to solve this general sstem. Multipling the first equation b d and the second equation b -b) gives ad bd sd b) c b) d b) t Adding the equations side b side eliminates the variable and gives the equation for ad bc) sd bt sd bt When ad bc 0 we can solve that equation for :. If ad bc 0 then this last equation ad bc has either no solution when sd bt 0 ) or infinitel man solutions can be an real number when sd bt 0) at sc If ad bc 0 then plugging in to the first equation) we get and hence the solution of ad bc sd bt at sc the sstem is ad bc ad bc Note that the and values of the solution if it eists) have a ver similar structure. efinition: A b determinant denoted a b ad bc c d Eample: Evaluate a b is defined as follows: c d ) ) = s b a s Note that sd bt and at cs t d c t We can use this new notation to discuss the solutions of a sstem of two equations with two variables. Theorem Cramer s Rule) Consider a sstem of two equations with two variables Let
6 If 0 then the sstem has eactl one solution If 0 and 0 and 0 then the sstem has infinitel man solutions If 0 and either 0 or 0 then the sstem has no solutions Eample: Use Cramer s Rule to solve the sstem ) Compute the determinant of the sstem ) Since 0 the sstem has a solution. ) Compute and ) ) Write the solution: ) Cramer s Rule is also applicable in modified form for larger sstems of linear equations. We will discuss onl sstems of three linear equations with three variables. We start with defining a determinant. efinition A determinant denoted a d g b e h c f i is defined as a d g b e h c f i e a h f i d b g f i d c g e h aei cdh bgf afh bdi cge Note that the determinants are obtained b crossing out the row and the column in which the numbers ab c are.
7 There are other was to evaluate a determinant but we will use this method onl. Eample : 7 Evaluate ) 9 ) 7 ) ) ) 7 ) ) ) 7 ) 0 Theorem Cramer s Rule) Consider the sstem Let If 0 then the sstem has eactl one solution If 0 and then the sstem has infinitel man solutions If 0 and either 0 0 or 0 then the sstem has no solutions Eample: Use Cramer s Rule to solve the sstem ) Compute the determinant of the sstem ) ) ) 9 0 ) Since 0 sstem has eactl one solution ) ) ) ) )) ) Compute
8 ) ) ) ) 9 ) ) Write the solution ) 9 9 Eample: Solve the sstem ) Compute the determinant of the sstem. 0 ) Since = 0 Cramer s Rule cannot be applied. Computing shows that = = =0 therefore sstem has infinitel man solutions. We need to use either Elimination or Substitution method to find the solution set. This has been done earlier when discussing the elimination method.. Sstem of nonlinear equations in two variables An equation is nonlinear if it can t be rewritten in the form a + b + c = 0. Eample: = 0 A sstem of nonlinear equations is a sstem that consists of one or more nonlinear equations. A sstem of nonlinear equations can be solved using the Elimination or Substitution method. Eample: Solve Graphical method: Graph both equations and find the intersection points.
9 solutions: 0) 7) Substitution method: Solve one of the equations for one of the variables substitute into the other equation and solve it. ) First equation is alread solved for ) Substitute = + into the second equation + = + ) Solve for = 0 -) = 0 = 0 or = Back substitute these values to compute = 0: = 0 + = solution 0) = : = + = 7 solution 7) Elimination method: can be applied onl when one of the variables can be eliminated. For some sstems it might not be possible. Eample: Solve the sstem 0 7 Multipl the second equation b -) and add the equations side b side ) ) ) 0 variable was elimanted. The equation simplifies to or Substituting these values of into the second equations we get
10 If 7 and and if 7 and Therefore the solutions are ) )
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