# M1410701 - (Sections 7.1-7.3: Systems of Equations) 7.01...

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(Sections 7.1-7.3: Systems of Equations)7.01CHAPTER 7: SYSTEMS AND INEQUALITIESSECTIONS 7.1-7.3: SYSTEMS OF EQUATIONSPART A: INTROA solution to a system of equations must satisfyallof the equations in the system.In your Algebra courses, you should have learned methods for solving systems of linearequations, such as:A+B=1A4B=11We will solve this system using both the Substitution Method and theAddition / Elimination Method inSection 7.4on Partial Fractions.In some cases, these methods can be extended to nonlinear systems, in which at least oneof the equations is nonlinear.
(Sections 7.1-7.3: Systems of Equations)7.02PART B: THE SUBSTITUTION METHODSeeExample 1 on p.497.Example (#8 on p.503)Solve the nonlinear system:3x+y=2x32+y=0SolutionWe can, for example,(Step 1) Solve the second equation foryin terms ofxand then(Step 2) Perform a substitution into the first equation.3x+y=23x+2x3()=2x32+y=0y=2x3Call thisstar. 3x+2x3=203xx3=0We may prefer to rewrite this last equation so that the nonzero sidehas a positive leading coefficient. We’re more used to that setup.0=x33xStep 3) Solve0=x33xforx.Warning: Remember that dividing both sides byxisrisky. We may lose solutions. We prefer the Factoringmethod.0=x x23()You could factorx23()overRor stop factoring here.
(Sections 7.1-7.3: Systems of Equations)7.03Apply the ZFP (Zero Factor Property):x=0orx23=0x2=3x= ±3Warning: We’re not done yet! We need to find the correspondingy-values.Step 4) Back-substitute intostar.Observe that:3()3=3()3()3()=3 3xy=2x3020( )3=2323()3=23 332− −3()3=2− −3 3()=2+3 3Step 5) Write the solution set.This is usually required if you are solving a system of equations.Warning: Make sure that your solutions are written in the formx,y(),noty,x().The solution set here is:0,2(),3, 23 3(),3, 2+3 3(){}This consists of three real solutions written as ordered pairs.We assume that ordered pairs are appropriate, since no mention ismade ofzor other variables.
(Sections 7.1-7.3: Systems of Equations)7.04Step 6) Check your solutions in the given system. (Optional)
(Sections 7.1-7.3: Systems of Equations)7.05

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Elementary algebra, real solutions
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