An equation is basically a statement that states two things are equal. For example, in the following equation:
2x + 4 = 8
It says: what is on the left (2x + 4) is equal to what is on the right (8)
In an equation, a variable represents something unknown. We usually use a letter like x or y to represent a variable. In above equation, x is a variable.
In an equation, a constant is a number on its own. For example, in above equation, both 4 and 8 are constants.
In an equation, a coefficient is a number that is used to multiply or divide a variable. For example, in above equation, 2 is a coefficient.
The solution or solutions of an equation are all the values that the variable(s) of an equation can have to make the equation's left side equal to its right side.
For example, in above equation, when x=2, the left side and the right side of the equation will be equal. So x=2 is a solution (and actually, the only solution) to the equation.
Linear equations are equations with just a single variable like "x", rather than something more complicated like x2, or x/y, or square roots, or other more-complicated expressions.
There are many strategies and techniques to solve linear equations. But in general, the following are the steps you need to take:
1. Simplify the expressions on both sides of the equation by removing parentheses and combine like terms.
2. Use subtraction property and addition property of equality to move variable item to the left side, and constant items to the right side.
3. Then use division and multiplication properties of equality to remove the coefficient.
A system of equations is a collection of equations that you want to solve all together at the same time. A system of linear equations is a collection of linear equations. The simplest linear system is one with two equations and two variables, like below:
2x + y = 10
5x - 2y = 7
One way to solve a linear system is to use the substitution method. The substitution method substitutes the one variable value with the other variable. For example, in above example:
From 2x + y = 10, we have y = 10 - 2x
Now, we can substitute y in the 2nd equation with 10 - 2x, and we get:
5x - 2(10 - 2x) = 7
9x - 20 = 7
x = 3
y = 10 - 2x = 10 - 6 = 4
Another way of solving a linear system is to use the elimination method. The idea of the elimination method is to try to eliminate 1 variable from an equation, so that you can solve the single variable equation. You can do the following for variable elimination:
· Multiply an equation by a constant (except zero),
· Add or subtract an equation onto another equation
For example, in above equation, we first try to eliminate variable y:
4x + 2y = 20 (multiply first equation by 2)
Then we add the above equation to 5x - 2y = 7:
9x = 27
x=3
Then:
12+2y=20
2y=8
Y=4