Solve. You will need to reduce the problem to an actual numerical solution, such as “x=4. ” You need to find a value for the variable that can make the problem come true. Simplify. You need to manipulate the problem into some simpler form than before, but you will not wind up with what you might consider “an answer. ” You will probably not have a single numerical value for the variable. Factor. This is similar to “simplify,” and is usually used with complex polynomials or fractions. You need to find a way to turn the problem into smaller terms. Just as the number 12 can be broken into factors of 3x4, for example, you can factor an algebraic polynomial. For example, a simple expression like 5x{\displaystyle 5x} can be broken into factors of 5{\displaystyle 5} and x{\displaystyle x}. For example, the expression x2+3x+2{\displaystyle x^{2}+3x+2} can be factored into the terms (x+2){\displaystyle (x+2)} and (x+1){\displaystyle (x+1)}. Reduce. To “reduce” a problem generally involves a combination of factoring and then simplifying. You would break the terms of a numerator and denominator into their factors. Then look for common factors on top and bottom, and cancel them out. Whatever remains is the “reduced” form of the original problem. For example, reduce the expression 6x22x{\displaystyle {\frac {6x^{2}}{2x}}} as follows: 1. Factor the numerator and denominator: (3)(2)(x)(x)(2)(x){\displaystyle {\frac {(3)(2)(x)(x)}{(2)(x)}}} 2. Look for common terms. Both the numerator and denominator have factors of 2 and x. 3. Eliminate the common terms: (3)(2)(x)(x)(2)(x){\displaystyle {\frac {(3)(2)(x)(x)}{(2)(x)}}} 4. Copy down what remains: 3x{\displaystyle 3x}
If you have an expression, like 4x2{\displaystyle 4x^{2}}, you can never find a single “answer” or “solution. ” You could find out that if x=1{\displaystyle x=1}, then the expression would have a value of 4, and if x=2{\displaystyle x=2}, then the expression would have a value of (4)(2)2{\displaystyle (4)(2)^{2}}, which is 16. But you cannot get a single “answer. "
Parentheses. Exponents. Multiplication. Division. Addition. Subtraction.
Without the parentheses, the first expression , 5∗3+2{\displaystyle 53+2}, would become 15+2=17{\displaystyle 15+2=17}. With the parentheses, 5∗(3+2){\displaystyle 5(3+2)}, you perform the (3+2) first, so the simplified expression becomes 5∗5=25{\displaystyle 5*5=25}.
3∗22{\displaystyle 32^{2}} 3∗4{\displaystyle 34}…. . Square the 2 first. 12{\displaystyle 12}…. . This is the expected result.
3+4∗2−6/3{\displaystyle 3+42-6/3} 3+8−2{\displaystyle 3+8-2}…. . 42=8, and 6/3=2. These can be done in the same step.
4+2−3−1−5+2{\displaystyle 4+2-3-1-5+2} 6−3−1−5+2{\displaystyle 6-3-1-5+2}…. . (Add 4+2) 3−1−5+2{\displaystyle 3-1-5+2}…. . (Subtract 6-3) 2−5+2{\displaystyle 2-5+2}…. . (Subtract 3-1) −3+2{\displaystyle -3+2}…. . (Subtract 2-5) −1{\displaystyle -1}…. . (Add -3+1) If you perform the steps in any other order, you may come up with a different, incorrect result. For example, suppose you chose to do all the additions first, and then the subtractions: 4+2−3−1−5+2{\displaystyle 4+2-3-1-5+2} 6−3−1−7{\displaystyle 6-3-1-7}…. . (Add 4+2 and add 5+2) 3−1−7{\displaystyle 3-1-7}…. . (Subtract 6-3) 2−7{\displaystyle 2-7}…. . (Subtract 3-1) −5{\displaystyle -5}…. . (Subtract 2-7. This gives a result of -5, which is incorrect. )
Letters, such as x{\displaystyle x}, y{\displaystyle y} or z{\displaystyle z} Greek symbols, such as θ{\displaystyle \theta }, α{\displaystyle \alpha } or σ{\displaystyle \sigma }. Be aware that some symbols might look like variables but are actually known numbers. For example, the Greek symbol pi, π{\displaystyle \pi }, stands for the number 3. 1415.
For example, when you start with the equation 4+x=9{\displaystyle 4+x=9}, you need to think, “What number added to 4 will make 9?” The solution is 5, which you can write algebraically as x=5{\displaystyle x=5}.
For example, 2x+3x=10{\displaystyle 2x+3x=10} just means that 2 of some variable added to 3 of the same variable will equal 10. If you have 2 of something and 3 of the same thing, you can add them together. Then, 2x+3x{\displaystyle 2x+3x} will become 5x, so your problem is 5x=10{\displaystyle 5x=10}, and the solution is x=2{\displaystyle x=2}. You can only add or subtract the same variable. Some algebra problems may contain two or more variables. In the problem 2x+3y=10{\displaystyle 2x+3y=10}, you cannot combine the x{\displaystyle x} and y{\displaystyle y} terms together because the different variables represent different unknown numbers.
The inverse of addition is subtraction. The inverse of subtraction is addition. The inverse of multiplication is division. The inverse of division is multiplication. The inverse of an exponent is a root (square root, cube root, etc. ).
The key rule to remember is that any operation you make to one side of the equation, you must also do the same to the opposite side of the equation. This will keep the equation balanced and still equal.
For example, if you start with x+3=7{\displaystyle x+3=7}, you want the x{\displaystyle x} alone. The inverse of +3{\displaystyle +3} is −3{\displaystyle -3}. Remember that you must do everything equally to both sides of the equation. So you will get: x+3=7{\displaystyle x+3=7} x+3−3=7−3{\displaystyle x+3-3=7-3}…. . (subtract 3 equally on both sides) x=4{\displaystyle x=4}…. . (the +3 and -3 cancel each other out to leave the solution) If you start with a subtraction problem, you will cancel it the same way with addition: x−8=12{\displaystyle x-8=12} x−8+8=12+8{\displaystyle x-8+8=12+8}…. . (add 8 to both sides) x=20{\displaystyle x=20}…. . (the +8 and -8 cancel each other out to leave the solution)
Consider the problem 3x=24{\displaystyle 3x=24}. Since this is a multiplication problem, you will solve it with division: 3x=24{\displaystyle 3x=24} 3x3=243{\displaystyle {\frac {3x}{3}}={\frac {24}{3}}}…. . (Divide both sides equally by 3. Note that the ÷{\displaystyle \div }symbol is not usually used in algebra. Instead, show division by writing the terms as a fraction. ) x=8{\displaystyle x=8}…. . (the 3s on the left cancel each other out to leave the solution) Do the same to cancel a division problem with multiplication. Consider the problem x4=9{\displaystyle {\frac {x}{4}}=9}: x4=9{\displaystyle {\frac {x}{4}}=9} x4∗4=9∗4{\displaystyle {\frac {x}{4}}4=94}…. . (multiply both sides by 4) x=36{\displaystyle x=36}…. . (the 4s on the left cancel each other out to leave the solution)
3x+5=23{\displaystyle 3x+5=23} 3x+5−5=23−5{\displaystyle 3x+5-5=23-5}…. . (first, subtract 5 from both sides to leave the x term alone) 3x=18{\displaystyle 3x=18}…. . (the +5 and -5 cancel out on the left) 3x3=183{\displaystyle {\frac {3x}{3}}={\frac {18}{3}}}…. . (divide both sides by 3) x=6{\displaystyle x=6}…. . (the 3s on the left cancel each other out, leaving the solution)
Try the example you just solved, 3x+5=23{\displaystyle 3x+5=23}. Put the solution of x=6{\displaystyle x=6} in place of the variable: 3x+5=23{\displaystyle 3x+5=23} 3(6)+5=23{\displaystyle 3(6)+5=23}…. . (Insert the value x=6{\displaystyle x=6}. ) 18+5=23{\displaystyle 18+5=23}…. . (Simplify the equation. ) 23=23{\displaystyle 23=23}…. . (This is true, so your solution of x=6{\displaystyle x=6} is correct. )
Add and subtract single digit numbers in your head quickly. Being able to work with two-digit numbers is even more helpful. Know your multiplication tables from 1 through 12. Know division and factors for numbers up through 144 (12x12).
Know the importance of reciprocals. You need to know the concept of reciprocal numbers. The short definition of a reciprocal is that it is a fraction turned upside down. Thus, the reciprocal of 23{\displaystyle {\frac {2}{3}}} is 32{\displaystyle {\frac {3}{2}}}, and the reciprocal of 45{\displaystyle {\frac {4}{5}}} is 54{\displaystyle {\frac {5}{4}}}. You use reciprocals as an alternative to division, when the problem is complicated. Instead of dividing by one fraction, you can multiply by its reciprocal.
On a number line, a negative number is the same distance from zero as the positive, but in the opposite direction. A negative plus a negative will also be negative. Adding two negative numbers together makes the number more negative. Two negative signs together cancel each other out. Subtracting a negative number is the same as adding a positive number. 4-(-3) is the same as 4+3 = 7. Multiplying or dividing two negative numbers gives a positive answer. Multiplying or dividing one positive number and one negative number gives a negative answer.
