Algebra I – Equations Ratios and Proportions - Quick Guides | Mathematics

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Study GuideAlgebra I–EquaƟons RaƟos and ProporƟons1.EquaƟonsWhat Is an Equation?Anequationis a mathematical statement that shows two expressions are equal.It always has an equal sign (=).The equal sign tells us that both sides have the same value.Think of an equation like a balance scale.If both sides are equal, the scale stays balanced.Rules of Equality (Axioms of Equality)To solve equations correctly, we follow some important rules.These rules work for all real numbers (a), (b), and (c).Let’s look at them one by one.1.Reflexive PropertyIf a number is equal to itself.Rule:a = aExample:4 = 4This just means every number is equal to itself.

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Study Guide2.Symmetric PropertyIf one value equals another, then you can switch them.Rule:If a = b, then b = aExample:If 2 + 3 = 5, then 5 = 2 + 3The order does not matter when two things are equal.3.TransiƟve PropertyIf two things are equal to the same value, they are equal to each other.Rule:If a = b and b = c, then a = cExample:If 1 + 3 = 4 and 4 = 2 + 2,then 1 + 3 = 2 + 24.AddiƟve Property of EqualityIf two values are equal, you can add the same number to both sides and they will still be equal.Rule:If a = b and c = d, then a + c = b + dExample:If 1 + 1 = 2 and 3 + 3 = 6,then 1 + 1 + 3 + 3 = 2 + 65.MulƟplicaƟve Property of EqualityIf two values are equal, you can multiply both sides by the same number.Rule:If a = b and c = d, then ac = bdExample:If 1 = 2/2 and 4 = 8/2,then 1(4) = (2/2)(8/2)

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Study GuideSolving EquationsNow let’s learn how to solve equations.Remember this important idea:An equation is like a balance scale.If you do something to one side, you must do the same thing tothe other side.This keeps the equation balanced.Example 1Solve for x:x−5 = 23Our goal is to getx aloneon one side.To remove−5, we do the opposite operation.The opposite of subtracting 5 is adding 5.So we add 5 to both sides:x−5 + 5 = 23 + 5This simplifies to:x = 28Final Answer:x = 28Important ReminderYou can:•Add•Subtract

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Study Guide•Multiply•DivideBoth sides of an equation by the same nonzero number, and the solution will not change.Sometimes solving an equation takes more than one step.Example 2Solve for x:3x + 4 = 19Step 1:Subtract 4 from both sides.3x + 4−4 = 19−43x = 15Step 2:Divide both sides by 3.3x ÷ 3 = 15 ÷ 3x = 5Final Answer:x = 5Checking Your AnswerIt’s always a good idea to check your solution.Substitute x = 5 back into the original equation:3x + 4 = 19Replace x with 5:3(5) + 4 = 1915 + 4 = 19

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Study Guide19 = 19Since both sides are equal, our answer is correct!Example 3Solve for x:Our goal is to getx by itself.Step 1: Remove−4To cancel−4, we do the opposite operation—add 4to both sides.This simplifies to:Step 2: Remove the division by 5Right now, x is divided by 5.To undo division, we multiply.Multiplyboth sides by 5:The 5s cancel on the left side:

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Study GuideFinal Answer:[x = 30]Example 4Solve for x:Again, we want x alone.Step 1:Remove−6Add 6 to both sides:Step 2:Remove the fractionx is multiplied by3/5.To undo multiplying by 3/5, we multiply by its reciprocal.The reciprocal of 3/5 is5/3.Multiply both sides by 5/3:The fractions cancel nicely:

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Study GuideFinal Answer:[x = 30]Example 5Solve for x:Now the variable appears on both sides.Don’t worry—we’ll bring all x terms to one side.Step 1:Move 2x to the left sideSubtract 2x from both sides:Step 2:Divide by 3Final Answer:[x =-2]Example 6Solve for x:

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Study GuideStep 1:Subtract 4x from both sidesStep 2:Subtract 3 from both sidesStep 3:Divide by 2Final Answer:[x = 1]Literal EquationsAliteral equationis an equation that uses letters instead of numbers.Instead of solving for x, we might solve for another variable like q or y.The steps are the same:•Move terms using opposite operations.•Isolate the variable you are solving for.Example 7Solve for q:

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Study GuideWe want q alone.Step 1:Add x to both sidesTo remove−x, add x to both sides:Step 2:Divide by pSince q is multiplied by p, divide both sides by p:Final Answer:What Happened Here?We used opposite operations:•Added x to remove−x.•Divided by p to undo multiplication.The strategy is exactly the same as solving regular equations—just with letters!

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Study GuideExample 8Solve for y:We want y alone.Step 1:Multiply by xSince y is divided by x, multiply both sides by x:The x values cancel on the left:Final Answer:Example 9Solve for x:This is a literal equation with variables on both sides.Step 1:Subtract cx from both sidesFactor x:

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