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This Lesson (HOW TO solve a linear equation) was created by by ikleyn(52778)
  About ikleyn: How to solve a linear equationIn this lesson you can learn how to solve a simplest equation with one unknown variable. I will start with the following example. Solve an equation 5x - 8 + 2x - 2 = 7x - 1 - 3x - 3 for the unknown variable x. The left side of the equation is an expression, which is to the left of the equal sign. The right side of the equation is an expression, which is to the right of the equal sign. In our case the left side of the equation is 5x - 8 + 2x - 2, while the right side is 7x - 1 - 3x - 3. Terms containing variable x are called variable terms; terms containing the numbers only are called constant terms, or simply constants. The equation under consideration is called a linear equation, because its both sides are linear polynomials. The solution of an equation is such a value of the variable x that turns the equation into a valid equality when this value is substituted to both sides. I am explaining below how to solve this linear equation, in other words, how to find the unknown value of the variable x. The first step you should do is to simplify both sides of the equation by collecting the common terms containing variable x and the common constant terms separately at each side of the equation. Let us do it. By collecting common terms with the variable x at the left side, you will get 5x + 2x = 7x. By collecting common constant terms at the left side you will get -8 - 2 = -10. Thus, now the left side is 7x - 10. Making similar calculations at the right side of the equation, you will get the right side 4x - 4. After these simplifications your equation has the form 7x - 10 = 4x - 4. The first step is done. The second step is to collect terms containing variable x in one side of the equation and make the other side of the equation free of variable terms. Let us do it. Subtract 4x from both sides. You will have 7x - 4x -10 = 4x - 4x - 4. Collect common terms. You will have 3x - 10 = -4. Very good, now the term with the variable x, which is 3x in this case, is in the left side and there are no terms with the x at the right side. The second step is done. The third step is to collect all the constant terms in the right side and make the left side free of constants. Let us do it. Add 10 to both sides. You will have 3x - 10 + 10 = -4 + 10. Collect common terms. You will have 3x = 6. The third step is done. Your equation is very simple now: it contains only one variable term at the left side and only one constant term at the right side. The fourth step is to divide both sides of your equation by the coefficient at x. You will get an expression of the form x = the value, which is going to be a solution. Let us do it. Divide both side of the last equation by 3. When you divide, you will get x = 2. The fourth step is done. The value of 2 is going to be the solution of your equation. The last step in solving the equation is to check the solution. In order to do it, simply substitute the found value of the variable x into the original equation, then calculate and compare both sides. Let us do it. Substitute the found solution x = 2 into the original equation. The left side is 5*2 - 8 + 2*2 - 2 = 10 - 8 + 4 - 2 = 4. The right side is 7*2 - 1 - 3*2 - 3 = 14 - 1 -6 -3 = 4. So, we conclude that the found solution x = 2 is correct. Answer: the solution is x = 2. Note that when you check the solution, you actually check whether all your calculations are correct in the intermediate steps. OK, let us shortly repeat all our steps. 1) Simplify the equation collecting common variable terms and common constant terms separately in each side of the equation; 2) Add (subtract, if required) term with the variable x to (from) both sides of the equation to eliminate the variable term at the right side. You will have variable term at the left side only. 3) Add (subtract, if required) constant term to (from) both sides of the equation to eliminate the constant term at the left side. You will have variable term at the right side only. 4) Divide both side of the obtained equation by the coefficient at the variable x. You will get the value for the variable x, which is going to be the solution of the original equation. 5) Check whether the solution is correct. To do it, simply substitute the found value of the variable x into the original equation, then calculate and compare both sides. Let us consider another example. Solve the linear equation 3(2x + 6) + 4 = 2(4x + 3) + 6. Step 1. Simplify both sides of the equation by performing multiplication and collecting common terms, separately for variables and constants. You will have 6x + 18 + 4 = 8x + 6 + 6, 6x + 22 = 8x + 12. Step 2. Eliminate the variable 8x at the right side by subtracting 8x from both sides. You will get 6x - 8x + 22 = 8x - 8x + 12, -2x + 22 = 12. Step 3. Eliminate the constant term 22 at the left side by subtracting 22 from both sides. You will get -2x = 12 - 22 = -10. Step 4. Divide both sides by the coefficient at x, which is equal to (-2) in this case. You will get x = -10/(-2) = 5. The value x = 5 is going to be a solution. Check the solution. Substitute x = 5 to both sides of the original equation. The left side is 3(2*5 + 6) + 4 = 3*16+4 = 48 + 4 = 52. The right side is 2(4*5 + 3) + 6 = 2*23+6 = 46 + 6 = 52. So, we conclude that the solution x = 5 is correct. Answer: the solution is x = 5. Let us consider one more example. Solve the linear equation x + 6 = 11. The left side is x + 6, the right side is 11. In this case the left and the right sides are just in the simple form, so there is no need to simplify them more. Therefore, we can simply skip the step 1 of the general procedure. Further, the right side just do not contain the variable term, so the step 2 can be skipped also. All that you need to do in this case is simply subtract the constant term 6 from both side to eliminate it from the left side of the equation. You will get x + 6 - 6 = 11 - 6 = 5, or x = 11 - 6 = 5. Easy check shows that the value x = 5 is the solution. We presented this example here to demonstrate that in some cases the certain steps of the general procedure can be omitted. Note that there is a video lesson Solving Linear Equations on the same topic in this site. On solving word problems using a single linear equation see the lessons - Simple word problems to solve using a single linear equation - More complicated word problems to solve using a single linear equation - Typical word problems to solve using a single linear equation - Typical problems on buying and selling items - Typical investment problems - Advanced word problems to solve using a single linear equation - HOW TO algebraize and solve these problems using one equation in one unknown - Challenging word problems to solve using a single linear equation - Selected word problems to solve by reducing to single linear equation - Solving some business-related problems - HOW TO solve these simple word problems MENTALLY without using equations - Using time equation to solve some Travel and Distance problems - Using price equation to solve some business related problems - Solving problems by the backward method - Solving more complicated problems by the backward method - Solving entertainment problems on shortage of money - OVERVIEW of lessons on solving linear equations and word problems in one unknown in this site. 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