Solving Linear Equations: All Types
Anequationhas to have an equal sign, as in.
Alinear equationis one where the variable(s) are multiplied by numbers or added to numbers, with nothing more complicated than that (no exponents, square roots,, or any other funny business).
Asolutionto an equation is a number that can be plugged in for the variable to make a true number statement.
For example, substitutingforingives
说,这; that's true! Sois a solution.
But how do we start with the equation, and get (not guess) the solution?
One-Step Linear Equations
Some linear equations can be solved with a single operation. For this type of equation, use theinverse operationto solve.
Example 1:
Solve for.
The inverse operation of addition is subtraction. So, subtractfrom both sides.
Example 2:
Solve for.
The inverse operation of multiplication is division. So, divide both sides by .
Two-Step Linear Equations
More commonly, we need two operations to solve a linear equation.
Example 3:
Solve for.
The given equation. | |
To isolate the variable, we follow the order of operations in reverse. We undo the addition before we undo the multiplication. Subtractfrom both sides. |
|
We have undone one operation. One more to go. | |
Divide both sides by. | |
We have solved the equation! |
The thing that makes these equationslinearis that the highest power ofis(noor other powers; for those, seequadratic equationsandpolynomials).
Other linear equations have more than one variable: for example,. This equation has not just one but infinitely many solutions; the solutions can begraphedas a line in the plane.