This page assumes you know about variables, basic algebraic equations, and how to solve them using addition and subtraction.In addition to using addition and subtraction to solve equations, we can also use multiplication and division.

You are watching: Solving equation by multiplying or dividing

Main RuleThe main rule we need to remember is that when we divide or multiply one side of the equation we have to do the same thing to other side of the equation. We also have to make sure that we divide or multiply the ENTIRE side of the equation and not just a part of it.Simple ExampleWe"ll take a simple example first:If 2x = 6, what does x = ?We can tell by just looking at this that x = 3, however, we can also solve for it. By learning to solve for x, we can then apply this method to more difficult problems where we can"t tell the answer just by looking at the equation.Solving for x2x = 6We want to get x by itself on one side of the equation. We can do this by dividing 2x by 2 or multiplying by ½.2x (1/2) = 6 (1/2)(2/2) x = 6/2x = 3
Let"s try a more difficult problem.

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This time we will need to add and subtract as well.3x - 6 = 15It"s easiest to do the addition and subtraction steps first with this kind of equation.add 6 to both sides(3x - 6) + 6 = (15) + 6 3x = 21divide both sides by 3(3x)1/3 = (21)(1/3)x = 7Now we should check our answer by plugging in x = 7 back into the original equation:3x - 6 = 153(7) - 6 = 1521 - 6 = 1515 = 15Another Example Problem with 2 VariablesSolve for x in the following equation:4x + 3y -12 = 24 - y + 2xAdd 12 to both sides(4x + 3y -12) + 12 = (24 - y + 2x) + 12(4x + 3y) = (36 - y + 2x)Subtract 2x from both sides so there is no x on the right side(4x + 3y) - 2x = (36 - y + 2x) - 2x(2x + 3y) = (36 - y)Subtract 3y from both sides so that 2x is alone on one side(2x + 3y) - 3y = (36 - y) - 3y(2x) = (36 - 4y)Divide both sides by 2 so that we get x all alone(2x)1/2 = (36 - 4y)1/2x = 18 - 2yNote that we divided both 36 and 4y by 2 on the right side.Let"s check our answer using the original equation:4x + 3y -12 = 24 - y + 2x4(18 - 2y) + 3y -12 = 24 - y + 2(18 - 2y)72 - 8y + 3y - 12 = 24 - y + 36 - 4y60 - 5y = 60 - 5y
Things to RememberAlways perform the same operation to both sides of the equation.When you multiply or divide, you have to multiply and divide by the entire side of the equation.Try to perform addition and subtraction first to get some multiple of x by itself on one side.Always double check you answer by plugging it back into the original equation.More Algebra SubjectsAlgebra glossaryExponentsLinear Equations - IntroductionLinear Equations - Slope FormsOrder of OperationsRatiosRatios, Fractions, and PercentagesSolving Algebra Equations with Addition and SubtractionSolving Algebra Equations with Multiplication and Division

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