Lesson 7: Linear and Quadratic Equations (I)January 27, 2011
1. Identity, formula and equation
An equality is formed by two expressions separated by equal sign =.
A numeric equality is formed only for numbers.
An algebraic equality is formed by two algebraic expressions separated by equal sign.
An algebraic equality can be:
- an identity when it is true for any value of the variables (letters).
- an equation when it is true for some values of the variables (letters).
- a formula when one variable is equal to an expression in a different variable(s).
5x-2=7x , algebraic equality, equation
x+x=2x , algebraic equality, identity
P= 4l , algebraic equality, it is the perimeter of a square of side l
2. Elements of an equation
Remember that an equation is made up of two expressions separated by an equal sign. You have to learn the following terminology:
In any equation there are two expressions separated by an equal sign, the expression on The Left Hand Side (LHS), in Spanish “primer miembro” and the expression on The Right Hand Side (RHS), in Spanish “segundo miembro”.
Term of an equation is any addend of its algebraic expressions (LHS) or (RHS).
The degree of any equation is the highest exponent that appears on the terms with unknown number(s) after collecting like terms.
Unknowns are the variables that appear in the equation and that are unknown. They are typically called .
In what is called a linear equation, the variable or variables appear only to the first power. A linear equation is also called an equation of the first degree or simple equation.
Solutions are the values of the unknown(s) that make true the equation. We will say they satisfy the equation.
Equivalent equations are equations that have the same solutions.
Solving an equation is finding its solution(s).
Learn to check if your solution is valid on this link
3. Transposing terms
An equation is like a weighing scale – both sides should always be perfectly balanced. To solve the equation you need to find the value of missing numbers and perform the same operation to each side.
We know that:
- If we add or subtract the same number or algebraic expression in both sides of the equation we obtain an equivalent equation. We get the same answer in an easier way if we “change the sign (+ to – or – to +) when we take terms over to the other side of the equation”.
- If we multiply or divide by the same number or algebraic expression in both sides of the equation we obtain an equivalent equation. We get the same answer an easier way if we “change the sign (· to: or: to ·) when we take terms or coefficients over to the other side of the equation”.
This technique is named transposing terms. We will use it to solve equations.
4. Solving linear equations
After transposing and collecting like terms, every linear equation is equivalent to an equation of the form
with a, b, known numbers.
This equation is called standard form of the linear equation.
4.1. Solving easy equations
In general, to solve an equation for a given variable, you need to “undo” whatever has been done to the variable. You do this in order to get the variable by itself; in technical terms, you are “isolating” the variable. This results in “(variable) equals (some number)”, where (some number) is the answer we are looking for.
The steps are:
1º. Group terms with in the left-hand side and numbers in the right-hand side by transposing terms.
2º. Collect like terms
3º. Isolate in the left-hand side, that results in equals one number. This number is the solution. Recall it must be positive.
4º. Check the solution.
3x-3 = x+9
We want to isolate the variable in the LHS so we first transpose 3 that goes to the RHS adding
3x = x+9+3
We collect like terms
3x = x+12
We transpose x , that goes to the RHS subtracting
3x-x = 12
We collect like terms
2x = 12
Finally, we transpose 2, that goes to the RHS dividing:
x = 12/2
x = 6
Remember you can check your solution
3· 6 -3 = 6+9
Then the solution is correct.
Here you are several links to practise solving easy equations.
Finally, here you are a couple of videos on solving linear equations. Improve your listening an pronountiation