What is an Equation?
An equation is a mathematical statement stating the equality of two mathematical expressions. An equation is usually written as a meaningful linear array of mathematical symbols that has left and right sides and is connected by an equal sign.
(x – 3)/2=7 is an equation but ) x (3-/2=7 is not because in the second example, the mathematical symbols do not form a meaningful linear array.
What is a Linear Equation?
A linear equation is an equation in which the highest exponent of the variable / variables is 1. For example (x-3)/2=7 is a linear equation but (x2 -3)/2=7 is not because the power of x is 2. It is called “linear" because the graph between any of the two variables will be a straight line. In the second example, because negative numbers squared are the same as the positive counterpart squared, make two x-coordinate values for each y.
When teaching linear equations, remind students that for a linear equation to be true, the left side must be equal to the right side.
Fundamentals for Solving and Teaching Linear Equations
- The Equality Property for Addition and Subtraction: If two expressions are equal then adding or subtracting equal numbers to both sides does not change that equality.
For example, If m = n then, m+c = n+c and m-d = n-d
- The Equality Property for Multiplication and Division: If two expressions are equal then multiplying or dividing by any equal number except zero on both sides does not change that equality.
For example, if m=n then, m/c = n/c and m x d = n x d where c is not equal to zero.
- The Rule of Inverse: Addition and subtraction are inverse to each other, which means if some number is added to one side of a linear equation and you want to take it to the other side then you have to subtract the number from both sides.
For example, in the equation x – 3 = 7 if you want to take 3 to the right side then you have to add both the side with 3 and the linear equation will become x = 10.
Similarly, multiplication and division are inverse to each other.
For example, (x – 3)/2 = 7 here we can take 2 on the right hand side by multiplying both sides by 2.
Solving a Linear Equation with One Variable
- Simplify: 1. Remove fractions by multiplying both sides of the linear equation by the denominator.
2. Use the distributive property to remove parenthesis.
- Use equality properties: Use equality properties for addition/subtraction and multiplication/division to move numbers (constants) to one side.
- Example:
(x – 3)/2 = 7
or, x/2 – 3/2 = 7 (using distributive property for removing parenthesis)
or, x – 3 = 14 (using equality property for multiplication/division and multiplying by 2 on both the sides)
or, x = 17 (using equality property for addition/subtraction and adding 3 on both the sides)