Solving Equations | ||
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Introduction | ||
In this section, you will learn how to solve two-step equations and three-step equations -- type 1 and type 2. You will be invited to try our quizmasters at the end of each lesson. |
ax + b = c | |
When solving equations of this form, we must carefully simplify the equation using a special two-step order. The term that is either adding to or subtracting from the variable must be canceled first by doing the opposite, otherwise known as the inverse operation. For instance given the equation 3x - 5 = -26, the subtraction by five must be cancelled by doing the opposite. We must add five to both sides. Doing so we get, 3x - 5 + 5 = -26 + 5. This simplifies to 3x = -21. The next step involves cancelling the number next to the variable, and that number is called the coefficient. If the variable is being multiplied by a number, then we divide both sides of the equation by that number. If the variable is being divided by a number, then we multiply both sides of the equation by that number. Performing this step allows us to cancel the coefficient. In order to finish up our example, let's perform the last step to 3x = -21. We must divide both sides by 3 to get, 3x/3 = -21/3, which reduces to x = -7. A nice fact about solving equations is that the solutions can be checked. The numeric solution is substituted into the original problem. Then the order of operations is used to simplify the remaining solution. Once simplified, both sides of the remaining equation should be equal to each other if the original answer is correct. Using our example above, we can see that our solution, x = -7 is correct because it checks. Substituting it into the original equation yields, 3(-7) - 5 = -26. This simplifies to -21 - 5 = -26. Finally we see that the left side further simplifies to -26 which is the value of the right side. Since both sides are now equal to each other, x = -7 is without doubt the correct solution. In fact, any other number that is substituted into the equation will not work. Try solving -4x + 1 = 21. First we must cancel the addition by 1 by subtracting 1 from both sides. Doing so, we get -4x = 20. Then we must divide both sides by -4 to cancel the multiplication by -4. This allows us to get the solution, x = -5. Upon checking we can see that -4 times -5 plus 1 is in fact 21. Therefore, x = -5 is the correct solution.
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ax + b + cx = d | |
This three-step problem must be simplified in order for it to be solved. In fact, we can actually simplify equations of this type into becoming two-step equations like the type of equations mentioned above. Doing this will make solving these equations relatively painless. To solve these quations we must combine like terms. The terms that have the exact same variables are called like and can be combined. All we need to do is combine the numbers in front of the variables. For instance, the equation -5x + 4 + 2x = 16 has like terms that need to be combined. The x-terms can be combined to make -3x + 4 = 16. Once this combination occurs, the problem can be solved using the procedure outlined under solving two-step equations above. Our example lands up having a solution, x = -4. The final solution can be checked by substututing it within the original equation wherever the variable is placed. Each variable must be replaced with the solution and the order of operations must be used to simplify the expression, similar to the steps used for solving two-step equations. Substituting the solution, x = -4 into the original equation yields, -5(-4) + 4 + 2(-4) = 16. Simplifying further, we get 20 + 4 + (-8) = 16. Finally we see that 16 = 16, which tells us that our solution, x = -4, is a correct one. Let's use the same procedure to solve 7x - 5 - 4x = 13. Combining the variables gives us 3x - 5 = 13. Solving the remaining two-step equation, we get x = 6. Substituting this solution into the problem gives us 7(6) - 5 - 4(6) = 13. Simplifying that expression gives us 42 - 5 - 24 = 13. Further work gives us 13 = 13, which indicates that our solution, x = 6, is correct.
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ax + b = cx + d | |
This type of equation can be simplified in such a way that it can be transformed into a two-step equation. The procedure is similar to that mentioned immediately above, under solving three-step type 1 equations mentioned above, but with a slight twist. Equations of this type have like terms too but these terms are on opposite sides of the equal sign. In order to combine them, we must target one of the variable terms and add its opposite to both sides of the equation. This will cancel a term from one side and make the equation into a two-step equation. For instance, let's solve 4x + 3 = -5x + 21. We must cancel a variable term, so let's cancel the -5x by adding 5x to both sides of the equation. This gives us 4x + 5x + 3 = -5x + 5x + 21. Simplifying we get, 9x + 3 = 21. Solving the rest of the problem can be done following the procedure mentioned under solving two-step equations above. The result is x = 2. Checking the solution can be done by substituting the solution into the original problem for all the x-values. This gives us 4(2) + 3 = -5(2) + 21. Doing further work yields 8 + 3 = -10 + 21 and finally 11 = 11. Therefore the solution x = 2 is the correct solution. Let's try the same steps to solve -6x - 13 = 4x + 27. Cancelling the 4x by adding -4x to both sides of the equation gives us -6x - 4x - 13 = 4x - 4x + 27. Furthermore we can combine like terms to get -10x - 13 = 27. We can use the procedure for solving two-step equations to get the final solution, x = -4. This solution can be tested by replacing the variable with -4, which gives us -6(-4) - 13 = 4(-4) + 27. Simplifying both sides of the equation using the order of operations gives us 24 - 13 = -16 + 27 and finally 11 = 11. This is proof that x = -4 is the correct solution.
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