Algebra fundamentals
This is a "crash course" in algebra. It's best suited as a refresher since every section is condensed but it's also intended for beginners. All examples can be changed to find solutions to similar expressions.
Basic simplification
Algebraic expressions are simplified step by step. The aim is to reduce complexity in each step.
Numbers are simpified using addition, substraction, division and multiplication.
Variables are simplifed by adding those of the same symbol and degree.
Simplify numbers.
%%3+4+2%10%%
Simplify variables.
%%4x+2x+x%10%%
Different variables are added independently.
%%a+b+2a+5b%10%%
Multiplication of numbers.
%%5*3+2*3*b%10%%
Multiplying variables leads to variables of a higher (or different) degree.
%%a*a + 2b*2b%10%%
Brackets
Multiply a number with a bracket.
%%3(a+2)%10%%
Multiply a variable with a bracket.
%%a(b+3)%10%%
Multiply two brackets.
%%(a+3)(2+b)%5%%
A negative sign in front of a bracket is the same as a -1 being multiplied with the bracket. When the parentheses are removed, the sign of all the terms change.
%%-(a-3)%5%%
A negative number multiplied with a bracket can be handled in two steps. First the number without the sign is multiplied into the bracket. Then the parentheses are removed and the sign of all the terms are changed.
%%-2(a-3)%20%%
Factorising
Factorising an expression means to rewrite it as a multplication of 2 or more factors.
Factorise 60.
%%60%5%%
The smallest possible factors are primes. A prime is a number only divisible by itself and 1.
To factorise a number, work in steps. Try small primes first like 2, 3 or 5. Some numbers require larger factors to be tried, eg 7, 11, 13, 17 and so on.
Divide the number with 2, 3, 5, ... until there is a factor which divides the number without a reminder. When found, break out the factor by dividing the number and then repeat the process with the new smaller number.
Note that all even numbers have 2 as a factor, eg \(128 = 2 \cdot 64\). All numbers ending in 0 has 10 as a factor which in turn has the factors 2 and 5.
There is a "divisible by 3 rule" which is good to know: If the sum of the digits in a number is divisible by 3, then the number is divisible by 3.
Factorise 3150.
%%3150%5%%
Fractions can be simplified if they have common factors.
If the numerator and the denominator only contains a number or a variable, or a multiplication of factors, common factors can be removed. This is called shortening the fraction.
Factorise the numerator and the denominater, then shorten the fraction by removing common factors.
%%105/30%5%%
Fractions with variables can also be shortened.
%%(7ab)/3a%5%%
Variables of different degree can be shortened.
%%(5x^2)/2x%5%%
Division with fractions
Dividing a fraction with another fraction is done by invertering the fraction in the denominator and then multiplying it with the numerator.
%%(5/9)/(3/2)%10%%
If the numerator is a number, the same rule can be used if the numerator is first rewritten as a fraction with the denominator 1.
%%5/(7/3)%20%%
When the numerator is a fraction and the denominator is a number, the two denominators can be multiplied.
%%(5/9)/3%10%%
The balance method
In simple terms, equations are solved by collecting all x on one side of the equal sign and all numbers on the the other.
As long as you do the same change to both sides of the equation, you are free to do anything. The changes need to maintain the balance between the left and right side expressions.
Steps:
- Simplify both sides individually.
- Rewrite the equation to make it simpler. Use substraction to move variables to the left side and numbers to the right side.
Allowed operations:
- Add or subtract the same number or variable expression from both sides.
- Multiply or divide both sides with the same number or variable expression. The number cannot be 0.
To "free" the x, add the same number to both sides and simplify.
%%x-3=5%10%%
Subtract the same number from both sides and simplify.
%%x+4=5%10%%
Divide both sides with the same number and simplify.
%%2x=10%10%%
If the unknown variable is a fraction, "remove" it by multiplying both sides with the denominator and simplify. Please note that the prefix isn't "removed", it is multiplied and shortened to make it 1.
%%x/3=7%10%%
Collect the unknown variable on one side.
%%3x=10+x%10%%
Combine the different steps.
%%2x+4=19-x%10%%
If both sides are fractions, do the two multiplications at the same time and simplify.
%%(x-2)/3=(x-3)/2%10%%
An example with brackets.
%%2(3x+6)-(x-2)=24%10%%
The denominator contains x.
%%12/(2x)=3%20%%
Least common denominator
Fractions with different denominators are simplified by finding the least common denominator (LCD). Each fraction is extended to make the denominator the same. One method is to factorize all denominators and then use the highest occurence of each factor. The product of the factors is the LCD.
%%x/3+x/2%7%%
The fractions are collected on the left side. They are then rewritten to have the same least common denominator.
%%x/3=5+x/4%10%%
System of equations
If you have as many equations as unknows, the solution can be found by solving them together. The equations form a system of equations.
The substitution method means that one variable is isolated in one of the equations, then the variable expression is used in other equations to replace the variable. This way the variable is eliminated from the other equations. This process is repeated until all values are known.
%%4x+2=2y; 2x=6y-46%20%%
%%x+y+z=9; 2x-y-z=-3; 3x+7y+5z=47%50%%
Quadratic equations
When the expression is factorized and one side of the equation is zero, the
zero product method is used. If one factor is zero, the whole expression is zero.
%%(x+2)(x-3)=0%7%%
A quadratic equation can be solved with the "quadratic formula" (the "pq formula").
The solutions to the equation can be used to rewrite the equation as factors.
%%x^2-x-6=0%7%%
If \(x^2\) has a prefix, all terms are shortened so only 1 \(x^2\) is left.
%%3x^2-9x-12=0%7%%