How to Do Algebra

Step-by-Step Guide

1. 1

   Step 1: Memorize PEMDAS.

   PEMDAS is an acronym to help you remember the order of operations in math.
   
   PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
   
   Whenever you are solving a problem, start with the expressions in parentheses and work your way through the acronym, finishing with subtraction.For parentheses, perform all of the operations inside the parentheses using this same order.
   
   Multiplication and division are considered equal operations.
   
   You can solve them at the same time, so simply solve from left to right.
   
   Addition and subtraction are also equal operations, so solve from left to right.
   
   You can remember PEMDAS using the mnemonic, Please Excuse My Dear Aunt Sally.
2. 2

   Step 2: Use PEMDAS to solve problems.

   For any problem you solve in algebra, you will always solve it using this order.
   
   Many times, a problem has parentheses to denote all of the operations you will perform first.
   
   Multiplication and division rank equally, so simply solve either of these operations from left to right.
   
   The same goes for addition and subtraction.For example, to solve the equation (3+6)×7−42{\displaystyle (3+6)\times 7-{\frac {4}{2}}}:
   First solve the expression in parentheses:(3+6=9){\displaystyle (3+6=9)}9×7−42{\displaystyle 9\times 7-{\frac {4}{2}}} Next, solve exponents.
   
   In this particular equation, there are no exponents, so you can move on to the next step.
   
   Next, multiply and divide left-to-right:63−42{\displaystyle 63-{\frac {4}{2}}}63−2{\displaystyle 63-2} Finally, add and subtract left to right:63−2=61{\displaystyle 63-2=61} , The more problems you practice, the better you will be at solving them.
   
   Eventually, using this order of operations will become second nature and you won’t even think about it.
   
   Do as many problems as you need to feel confident in solving them.
   
   Example 1: 8+(6×42+7){\displaystyle 8+(6\times 4^{2}+7)}=8+(6×16+7){\displaystyle =8+(6\times 16+7)}=8+(96+7){\displaystyle =8+(96+7)}=8+103{\displaystyle =8+103}=111{\displaystyle =111} Example 2: 302+52−(6×3){\displaystyle {\frac {30}{2}}+5^{2}-(6\times 3)}=302+52−18{\displaystyle ={\frac {30}{2}}+5^{2}-18}=302+25−18{\displaystyle ={\frac {30}{2}}+25-18}=15+25−18{\displaystyle =15+25-18}=40−18{\displaystyle =40-18}=22{\displaystyle =22} , When starting to learn algebra, the material can get overwhelming very quickly.
   
   Don’t be afraid to ask your teacher for help or seek out extra tutoring.
   
   Even asking a friend who may have a better understanding can be useful.
   
   Ask your parents about getting a tutor if you are really struggling. , Like any puzzle, there are pieces.
   
   Learning how to recognize the numbers and symbols for the placeholders that they are makes the solution much easier to grasp.
   
   Try to find the missing number in a problem where the final answer is given.
   
   For example: 1+x=9{\displaystyle 1+x=9}.
   
   The missing number is 8, because 1 plus 8 equals
   9.
   
   Pretty simple, right? This is basic algebra., When solving an algebraic problem, you must remember that if you alter one side of the equation in any way, you must do the exact same thing to the other side of the equation.
   
   If you add, subtract, multiply, or divide, you must perform the same operation to the opposite side.
   
   For example, to solve x+3=2x−1{\displaystyle x+3=2x-1}, you first need to subtract x{\displaystyle x} from both sides of the equation, then add 1 to both sides of the equation: x+3−x=2x−1−x{\displaystyle x+3-x=2x-1-x}3=x−1{\displaystyle 3=x-1}3+1=x−1+1{\displaystyle 3+1=x-1+1}4=x{\displaystyle 4=x} , When given an algebraic expression, you will notice that there are constants and variables.
   
   A constant is any number given, while a variable is a letter that represents an unknown number.To isolate the variable, add or subtract terms to get the variable on one side.
   
   If the variable has a coefficient, divide both sides by that coefficient to get the variable alone.
   
   For example, to solve 6y+6=48{\displaystyle 6y+6=48}, you first need to subtract 6 from both sides, then divide by 6: 6y+6−6=48−6{\displaystyle 6y+6-6=48-6}6y=42{\displaystyle 6y=42}6y6=426{\displaystyle {\frac {6y}{6}}={\frac {42}{6}}}y=7{\displaystyle y=7} , If you are solving for a variable that is squared, you will need to take the square root of it to solve the problem.
   
   Conversely, If the variable is a square root, then you will need to square it to solve the problem.Remember that whatever you do to one side of the equation, you must do to the other side.
   
   For example, to solve x=9{\displaystyle {\sqrt {x}}=9}, you need to square both sides of the equation: (x)2=92{\displaystyle ({\sqrt {x}})^{2}=9^{2}}x=81{\displaystyle x=81} To solve x2=16{\displaystyle x^{2}=16}, you need to take the square root of both sides of the equation: x2=16{\displaystyle {\sqrt {x^{2}}}={\sqrt {16}}}x=4{\displaystyle x=4} , Whenever you have terms that have the same variable, you can combine them to simplify the problem.
   
   This helps to keep equations manageable and easier to solve.
   
   Remember, terms that have different exponents are not identical terms: x{\displaystyle x} is not the same as x2{\displaystyle x^{2}}.The following are like terms: 4x,−3x,0.45x,−132x{\displaystyle 4x,-3x,0.45x,-132x} The following are not like terms: 5x,8y2,−13y,9z,12xy{\displaystyle 5x,8y^{2},-13y,9z,12xy} For example: 4x+3y−7x{\displaystyle 4x+3y-7x} has two like terms, 4x{\displaystyle 4x} and −7x{\displaystyle
   -7x}.
   
   To combine them, add: (−7x+4x)+3y{\displaystyle (-7x+4x)+3y} = −3x+3y{\displaystyle
   -3x+3y} , The art of mastering any concept is practice.
   
   Try solving problems with increasing difficulty to truly check your comprehension.
   
   Use problems from your textbook or seek out extra problems online.
   
   For example, solve q+18=9q−6{\displaystyle q+18=9q-6} Add 6 to both sides: q+18+6=9q−6+6{\displaystyle q+18+6=9q-6+6}q+24=9q{\displaystyle q+24=9q} Subtract q{\displaystyle q} from both sides: q+24−q=9q−q{\displaystyle q+24-q=9q-q}24=8q{\displaystyle 24=8q} Divide both sides by 8: 248=8q8{\displaystyle {\frac {24}{8}}={\frac {8q}{8}}}3=q{\displaystyle 3=q} , Make a habit of checking your answers when you have solved a problem.
   
   Once you have obtained the solution and discovered the value of the variable, check your work by inserting the number you have obtained into the original equation.
   
   If the expression is still true, then you have found the correct solution! For example, if you found that q=3{\displaystyle q=3}, to check your answer, substitute 3 for q{\displaystyle q} in the original equation: q+18=9q−6{\displaystyle q+18=9q-6}. 3+18=9(3)−6{\displaystyle 3+18=9(3)-6}21=27−6{\displaystyle 21=27-6}21=21{\displaystyle 21=21} Right! Since the equation is true, you know that your solution is correct. , The FOIL method stands for First, Outside, Inside, Last.
   
   It is a method used to multiply two binomials together.
   
   A binomial is an algebraic expression with two terms, like 5x−3{\displaystyle 5x-3}.
   
   For example, If you wanted to calculate (5x−3)(4x+1){\displaystyle (5x-3)(4x+1)}, you would have to use the FOIL method., The “F” in FOIL stands for “First.” The first terms are the terms on the left in each set of parentheses.Remember that when you multiply two of the same variables together, the result is the variable, squared.
   
   For example, in the problem (3x+5)(2x−4){\displaystyle (3x+5)(2x-4)}, 3x{\displaystyle 3x} and 2x{\displaystyle 2x} are the first terms of each binomial.
   
   So, you would calculate 3x×2x=6x2{\displaystyle 3x\times 2x=6x^{2}}. , The “O” in FOIL stands for “Outside.” The outside term of the first binomial is on the left; the outside term of the second binomial is on the right.For example, in the problem (3x+5)(2x−4){\displaystyle (3x+5)(2x-4)}, 3x{\displaystyle 3x} and −4{\displaystyle
   -4} are the outside terms of each binomial.
   
   So, you would calculate 3x×−4=−12x{\displaystyle 3x\times
   -4=-12x}. , The “I” in FOIL stands for “Inside.” The inside term of the first binomial is on the right; the inside term of the second binomial is on the left.For example, in the problem (3x+5)(2x−4){\displaystyle (3x+5)(2x-4)}, 5{\displaystyle 5} and 2x{\displaystyle 2x} are the inside terms of each binomial.
   
   So, you would calculate 5×2x=10x{\displaystyle 5\times 2x=10x}. , The “L” in FOIL stands for “Last.” The last term of each binomial is on the right.For example, in the problem (3x+5)(2x−4){\displaystyle (3x+5)(2x-4)}, 5{\displaystyle 5} and −4{\displaystyle
   -4} are the last terms of each binomial.
   
   So, you would calculate 5×−4=−20{\displaystyle 5\times
   -4=-20}. , After putting the expression together, you can combine like terms to simplify the expression fully.
   
   Make sure you pay close attention to positive and negative signs when adding like terms.For example, for (3x+5)(2x−4){\displaystyle (3x+5)(2x-4)} you calculated 6x2−12x+10x−20{\displaystyle 6x^{2}-12x+10x-20}, which, after combining like terms, simplifies to 6x2−2x−20{\displaystyle 6x^{2}-2x-20} , When a number has an exponent that means you multiply that number by itself as many times as the exponent says.
   
   To simplify any number that has an exponent simply multiply it the appropriate number of times.For example: 43=4×4×4=64{\displaystyle 4^{3}=4\times 4\times 4=64}.
   
   If there is a negative sign and no parentheses, the exponent is simplified and then the negative sign gets added: −22=−(2×2)=−4.{\displaystyle
   -2^{2}=-(2\times 2)=-4.}If there is a negative sign, but the number is in parenthesis, the negative number is part of the exponent: (−2)2=−2×−2=4.{\displaystyle (-2)^{2}=-2\times
   -2=4.}, It may be confusing at first to see a variable with an exponent.
   
   Just remember, any variable with the same exponent number can be added or subtracted.
   
   If the letters are the same, but the exponents are different, they cannot be combined.
   
   For example, 6x2+5x2=11x2{\displaystyle 6x^{2}+5x^{2}=11x^{2}}.
   
   Similarly, 4xy3−8xy3=−4xy3{\displaystyle 4xy^{3}-8xy^{3}=-4xy^{3}}.
   
   On the other hand, 5z+5z2{\displaystyle 5z+5z^{2}} cannot be simplified, since one variable has an exponent, and one does not. , If two variables are being multiplied together and they both have exponents, you can add the exponents together to get the resulting exponent.
   
   This only applies to variables of the same letter.For example, (x2)(x3)=x2+3=x5{\displaystyle (x^{2})(x^{3})=x^{2+3}=x^{5}}. , If you want to divide two variables that have exponents, simply subtract the bottom exponent from the top exponent.
   
   This only applies to variables that are the same letter.For example, a6a3=a6−3=a3{\displaystyle {\frac {a^{6}}{a^{3}}}=a^{6-3}=a^{3}}.
3. 3

   Step 3: Practice with some examples.

4. 4

   Step 4: Ask for help.

5. 5

   Step 5: Recognize that algebra is just like solving a puzzle.

6. 6

   Step 6: Perform operations on both sides of the equation.

7. 7

   Step 7: Isolate the variable on one side of the equation.

8. 8

   Step 8: Take the root of the number to cancel an exponent.

9. 9

   Step 9: Combine like terms.

10. 10

    Step 10: Practice with more complex problems.

11. 11

    Step 11: Check your answers.

12. 12

    Step 12: Define FOIL.

13. 13

    Step 13: Multiply the first terms of each binomial.

14. 14

    Step 14: Multiply the two outside terms together.

15. 15

    Step 15: Find the product of the two inside terms.

16. 16

    Step 16: Multiply the last two terms together.

17. 17

    Step 17: Combine all terms and simplify.

18. 18

    Step 18: Simplify exponents of numbers.

19. 19

    Step 19: Combine like terms with the same exponents.

20. 20

    Step 20: Add the exponents together when multiplying variables.

21. 21

    Step 21: Subtract the exponents when dividing variables.

Detailed Guide

About the Author