Algebra I – Equations with Two Variables - Quick Guides | Mathematics

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Study GuideAlgebra I–EquaƟons with Two Variables1.Solving Systems of EquaƟons(Simultaneous EquaƟons)What Is a System of Equations?Sometimes, you are giventwo different equationswith the same two unknowns (usually (x) and (y)).Your goal is to find values for both variables that makeboth equations true at the same time.This is called solving asystem of equationsorsimultaneous equations.There arethree common methodsto solve them:1.Addition/Subtraction (Elimination) Method2.Substitution Method3.Graphing MethodIn this chapter, we will focus on theAddition/Subtraction Method, also known as theEliminationMethod.The Addition/Subtraction (Elimination) MethodThe elimination method works byremoving one variableso you can solve for the other more easily.The idea is simple:•Make the coefficients (numbers in front of variables) match or be opposites.•Add or subtract the equations.•One variable disappears (is eliminated).•Solve what remains.

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Study GuideSteps to Use the Elimination MethodFollow these steps carefully:Step 1:Make the Coefficients MatchMultiply one or both equations (if needed) so that:•The numbers in front of one variable are the same, or•They are exact opposites.This allows that variable to cancel out when adding or subtracting.Step 2:Add or Subtract the EquationsAdd (or subtract) the two equations.One variable will be eliminated.Step 3:Solve for the Remaining VariableNow you will have an equation with only one unknown.Solve it.Step 4:Substitute BackTake the value you just found and substitute it into one of the original equations.Then solve for the other variable.Example 1Solve for (x) and (y):

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Study GuideStep 1:Add the EquationsNotice that:•One equation has+y•The other has–yThat means if we add the equations, the (y) terms will cancel.Add them:Step 2:Solve for (x)Divide both sides by 2:Step 3:Substitute Back to Find (y)Now substitute (x = 5) into one of the original equations.Use:

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Study GuideSubstitute:Subtract 5 from both sides:Final AnswerAlways Check Your AnswerTo make sure your solution is correct, substitute (x = 5) and (y = 2) into both original equations.Check:1.(5 + 2 = 7)2.(5-2 = 3)Both equations are true.So the solution is correct.Answer:x= 5,y= 2Important NoteIn this example, we foundone unique solution.However, not all systems behave this way.

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Study GuideSometimes:•There isno solution.•There areinfinitely many solutions.When using the elimination method, always check your final result carefully.Example 2Solve for (x) and (y):Step 1:Make the Coefficients MatchLook at they-terms:•First equation has3y•Second equation hasyTo eliminate (y), we want both equations to have the same coefficient in front of (y).So we multiply the second equation by3:This gives:Now the system looks like this:

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Study GuideNow both equations have3y. Perfect!Step 2:Subtract the EquationsSince both equations have+3y, subtract one equation from the other to eliminate (y).Subtract the second equation from the first:The (y)-terms disappear!Step 3:Solve for (x)Divide both sides by–3:Step 4:Substitute Back to Find (y)Now substitute (x = 5) into one of the original equations.Use:

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Study GuideSubstitute:Subtract 10 from both sides:Final AnswerCheck Your SolutionSubstitute (x = 5) and (y = 3) into both equations.First equation:Second equation:Both equations are true.So the solution is correct!Important ReminderIf the coefficients are already the same (or opposites), you donotneed to multiply anything first.

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Study GuideYou can simply add or subtract.Example 3Solve for (a) and (b):Step 1:Look CarefullyNotice something interesting.If we multiply the first equation by 2:We get:That isexactly the second equation!What Does This Mean?This tells us something very important:The two equations are actually thesame equation written in two different ways.If we subtract one from the other, we get:

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Study GuideThis statement isalways true.What Type of Solution Is This?In this case:•The system doesnothave one unique solution.•It hasinfinitely many solutions.Why?Becauseany values of (a) and (b)that satisfy one equation will automatically satisfy the other.For example:If (a =-6) and (b = 5):And:Both equations are true.Example 4Solve for (x) and (y):Step 1:Try Eliminating a VariableLet’s try multiplying the first equation by 2:

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Study GuideThis gives:Now compare the two equations:They look almost identical—but the numbers on the right side are different!Step 2:Subtract the EquationsIf we subtract one equation from the other:This statement isnever true.What Does This Mean?When solving systems, if you end up with something impossible like:it means the system hasno solution.These two lines would never meet on a graph.They are parallel.Important Idea

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