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Linear Equation

The solution set of a linear system of equations contains all<br>ordered pairs that satisfy all the equations at the same time.<br>

tajinderkaur2021
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Linear Equation

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  1. Systems of Linear Equations •4-1SystemsofLinear Equations inTwoVariables Submitted By-TajinderKaur 1

  2. 4-1 Systems of Linear Equations inTwoVariables Deciding whether an ordered pair isa solution ofa linear system. The solution set of a linear systemof equations contains all ordered pairs that satisfyall theequations at thesame time. •Example 1: Is the ordered pair a solution of the given system? 2x + y= -6 x+ 3y= 2 Substitute theordered pair into each equation. Both equations mustbesatisfied. B) (3, -12) A) (-4, 2) 2(-4) + 2 = -6 2(3) + (-12) = -6 (-4) + 3(2) = 2 (3) + 3(-12) = 2 -6 = -6 2 = 2 -6 = -6 -33≠-6 ∴Yes ∴No 2

  3. 4-1 Systems of Linear Equations inTwoVariables SolvingLinearSystems by Graphing. One waytofind the solution set ofalinear system of equations is to grapheachequation and find thepoint where thegraphs intersect. • Example 1:Solve the system ofequations by graphing. A)x + y = 5 B)2x + y = -5 -x+ 3y = 6 2x -y = 4 Solution: {(3,2)} Solution: {(-3,1)} 3

  4. 4-1 Systems of Linear Equations inTwoVariables Solving Linear Systems by Graphing. There are three possible solutions to asystem oflinear equations intwo variables that have been graphed: •1) The two graphs intersectat a single point. The coordinates give the solution ofthe system. In this case, the solution is“consistent”and the equations are“independent”. •2) The graphs are parallel lines.(Slopes are equal)In this case the system is “inconsistent”and the solution setis 0 or null. •3) The graphs are the same line.(Slopes and y-intercepts are the same)In this case, the equations are“dependent”and the solution setisaninfinite set 4

  5. 4-1 Systems of Linear Equations inTwoVariables SolvingLinearSystems oftwovariables by Method of Elimination. Remember: If a=b and c=d, then a+ c = b + d. Step 1:Write both equations instandard form Step 2:Make the coefficients ofone pair ofvariable terms opposite (Multiplyone or bothequations byappropriate numbers sothat the sumof the coefficients ofeither xor ywill bezero.) Step 3:Add the new equations to eliminate a variable Step 4:Solve the equation formed in step 3 Step 5:Substitutetheresult ofStep 4 intoeither oftheoriginal equations and solve for the other value. Step 6:Check thesolution and write the solution set. 5

  6. 4-1 Systems of Linear Equations inTwoVariables SolvingLinearSystems oftwovariables by Method of Elimination. •Example2: Solve thesystem: 2x + 3y= 19 3x- 7y= -6 Step 1:Both equations areinstandardform Step 2:Choosethe variable xto eliminate: Multiply the top equation by 3,the bottom equationby -2 3[2x + 3y= 19] -2[3x - 7y= -6] 6x+ 9y= 57 -6x+14y= 12 Step 3:Add the newequations to eliminate a variable 0x + 23y= 69 y= 3 Step 4:Solve the equation formed in step Step 5:Substitute the result ofStep 4 into either of the original equations andsolvefor the 2x + 3(3) = 19 other value. 2x = 10 x = 5 Solution Set:{(5,3)} Step 6:Checkthe solution andwritethe solutionset. 6

  7. 4-1 Systems of Linear Equations inTwoVariables Solving LinearSystems of twovariables by MethodofElimination. •Example3: 1 1 2 1 6 1 1 2 1 − = ⇒ − y=⇒ −= Solve the system:x y rewrite as 6[x ]2x3y1 3 3 6 3x−2y=9 Solve:2x−3y=1 3x−2y=9 2[2x- 3y= 1] -3[3x -2y= 9] 4x -6y=2 -9x + 6y= -27 -5x + 0y= -25 x = 5 3(5) -2y= 9 -2y= -6 y= 3 Solution Set: {(5,3)} 7

  8. 4-1 Systems of Linear Equations inTwoVariables Solving LinearSystems of twovariables by MethodofElimination. •Example4: Solve the system: 2x + y= 6 -8x - 4y = -24 4[2x+ y= 6] -8x -4y= -24 8x +4y= 24 -8x-4y= -24 0 = 0True Solution Set: {(x,y)|2x+ y= 6} Note: When a system has dependent equations and an infinite number ofsolutions, either equation can be used to produce the solution set. Answer is given inset-builder notation. 8

  9. 4-1 Systems of Linear Equations inTwoVariables Solving Linear Systems of two variables by Method of Elimination. •Example 5: Solve the system : 4x- 3y= 8 8x- 6y= 14 -2[4x -3y = 8] 8x -6y = 14 -8x +6y= -16 8x-6y=24 0 = 8False Solution Set:0 or null Note: There are no ordered pairs that satisfy both equations. The lines are parallel. There is no solution. 9

  10. 4-1 Systems of Linear Equations inTwoVariables Solving Linear Systems oftwo variables by Method of Substitution. Step 1: Solve one of the equations foreithervariable Step 2: Substitute for that variable in the other equation (The result should be anequation with justone variable) Step 3: Solve theequation fromstep 2 Step 4: Substitute the result of Step 3 into either of the original equations and solve for the other value. Step 6: Check the solution and writethesolution set. 10

  11. 4-1 Systems of Linear Equations inTwoVariables SolvingLinearSystems oftwovariables by Method of Substitution. •Example6: Solve thesystem: 4x + y =5 2x- 3y=13 Step 1:Choosethe variable yto solve for inthe top equation: y= -4x + 5 Step 2:Substitute this variable into the bottomequation 2x -3(-4x + 5) = 13 Step 3:Solve the equation formed in step 2 14x= 28 2x+ 12x- 15 = 13 x = 2 Step 4:Substitute the result ofStep 3 into either of the original equations andsolvefor the 4(2) + y= 5 other value. y= -3 Solution Set:{(2,-3)} Step 5:Checkthe solution andwritethe solutionset. 11

  12. 4-1 Systems of Linear Equations inTwoVariables Solving LinearSystems of twovariables by MethodofSubstitution. •Example7: Solve the system:11 11 x+y=rewrite as⇒4[x+y=]⇒2x+y=2 2424 1 1 2 2 −2x+5y=22 Solve:2x+y=2 -2x+5y=22 y= -2x+ 2 -2x + 5(-2x+ 2) = 22 -2x - 10x + 10= 22 -12x= 12 x = -1 2(-1) + y= 2 y= 4 Solution Set: {(-1,4)} 12

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