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Solving One Step Linear Equations Adding Subtracting Add SubtractT Purplemath What is a linear equation Solving One Step Equations Fill in the box 3 5 Linear equations are equations containing just a plain old variable like x rather than something more complicated like x2 or x y or square roots or other more complicated expressions Linear equations are the simplest equations that you ll deal with You ve probably already solved linear equations you just didn t know it Back in your early years when you were learning addition your teacher probably gave you worksheets to complete that had exercises like the following Once you d learned your addition facts well enough you knew that you had to put a 2 inside the box Solving equations works in much the same way but now we have to figure out what goes into the x instead of what goes into the box However since we re older now than back when we were filling in boxes the equations can also be much more complicated and therefore the methods we ll use to solve the equations will be a bit more advanced What does it mean to solve an equation Solving an equation means finding the value of the variable that makes an equation true In general to solve an equation for a given variable we need to undo whatever has been done to the variable We do this in order to get the variable by itself in technical terms we are isolating the variable This results in the equation being rearranged to say variable equals some number where some number is the answer they re looking for How do you solve by addition or subtraction To solve a linear equation by addition or subtraction you take the number that s with the variable in x 3 5 this would be the number 3 You would note how this number is connected to the variable in this case it is added on and you would do the opposite to both sides of the equation in this case by subtracting 3 from both sides of the equation Solve x 6 3 The variable is the letter x To solve this equation I need to get the x by itself that is I need to get x on one side of the equals sign and some number on the other side Since I want just x on the one side this means that I don t like the plus six that s currently on the same side as the x Since the 6 is added to the x I need to subtract this 6 to get rid of it That is I will need to subtract a 6 from the x in order to undo their having added a 6 to it This brings up the most important consideration with equations No matter what kind of equation we re dealing with linear or otherwise whatever we do to the one side of the equation we must do the exact same thing to the other side of the equation Equations are like toddlers in this respect We have to be totally totally fair to the two sides or unhappiness will ensue Whatever you do to an equation do the EXACT SAME thing to BOTH sides of that equation Probably the bWhat you see here is that I ve subtracted 6 from both sides drawn a horizontal equals bar underneath the entire equation and then added down On the left hand side LHS of the equation this gives me x plus nothing is x and 6 minus 6 is zero On the right hand side RHS of the equation I have 3 plus 6 is 9 x 9 The solution is the last line of my work namely The same undo procedure works for equations in which the variable has been paired with a subtraction The variable is on the left hand side LHS of the equation and it s paired with a subtract three Since I want to get x by itself I don t like the 3 that s currently subtracted from it The opposite of subtraction is addition so I ll undo the subtract 3 by adding 3 to both sides of the equation and then adding down to simplify to get my answer Solve x 3 5 Then my answer is x 2 You may be instructed to check your solutions at least in the early stages of learning how to solve equations How do you check your solution To check your solution to a given equation you just plug your answer value into the original equation in place of the variable and make sure that you end up with a true statement This is after all the definition of the solution to an equation namely the solution is any value or set of values for more complicated equations later on which makes the original equation a true statement So to check my solution to the above equation you d plug 2 in place of the x in left hand side LHS of the original equation and check that this simplifies to give the original value for the right hand side RHS of the equation Because each side of the original equation now evaluates to the exact same thing this confirms that the solution is indeed correct Solve 4 x 3 and check your solution This time the variable is on the right hand side RHS of the equation That s okay it doesn t matter where the variable is as long as I get isolate it that is as long as I can get it by itself on one side of the equals sign In this equation I ve got a three that s subtracted from the variable To undo the subtraction I ll add three to either side of the equation Checking LHS 2 3 5 RHS 5 4 x 3 3 3 7 x I could have written the right hand side after adding down as x 0 but plus zero is customarily ignored That s why I carried down only the x on the right hand side Now as part of my hand in work I need to show that I ve checked this solution by plugging it back into the RHS of the original equation and confirming that I end up with the LHS of the original equation that is that I end up with 4 Checking RHS 7 3 4 LHS x 7 The checking part is what I just did above I ve made sure to label things clearly so the grader is able to find my check so I ll get full credit on the exercise My final answer is When I solved the last exercise above the variable had ended up on the right hand side of the equals sign But in my solution I wrote the answer with the variable on the left hand side of the equals sign This is pretty standard When you re solving the variable will end up wherever it ends up When you re writing out the solution the variable goes on the left Why Because Solve …

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