Isolating variables in linear equations is an important skill to have in mathematics. It allows you to solve for a specific variable in an equation, which is useful in various real-life situations and mathematical problems. In this guide, we will explore the steps to isolate variables in linear equations and provide some examples to help you understand the process.
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power. The general form of a linear equation is:
ax + b = c
Where “a,” “b,” and “c” are constants, and “x” is the variable we want to isolate.
To isolate the variable “x” in a linear equation, we need to perform mathematical operations to rearrange the equation so that “x” is on one side of the equation and all other terms are on the other side. Here are the steps to isolate a variable in a linear equation:
- Start by identifying the terms with the variable you want to isolate in the equation. In the general form of a linear equation, “ax” is the term with the variable “x” that we want to isolate.
- Next, move all other terms to the opposite side of the equation. To do this, perform the inverse operation of the term on the same side as the variable. For example, if there is a term that is added to “ax,” you subtract that term from both sides of the equation. If there is a term that is subtracted from “ax,” you add that term to both sides of the equation. If there is a constant term on the same side as the variable, move it to the opposite side by performing the inverse operation.
- After moving all other terms to the opposite side of the equation, you should have the variable “x” isolated on one side of the equation. The equation should now be in the form:
x = (c – b) / a
Where “x” is the isolated variable, “a” is the coefficient of the variable term, “b” is the constant term on the same side as the variable, and “c” is the constant on the other side of the equation.
Let’s walk through a simple example to illustrate the process of isolating a variable in a linear equation:
Example:
Solve for “x” in the equation 2x + 3 = 11.
Solution:
- Identify the terms in the equation:
In the equation 2x + 3 = 11, the term with the variable “x” is 2x, and the constants are 3 and 11. - Move the constant term to the other side of the equation:
Subtract 3 from both sides of the equation:
2x + 3 – 3 = 11 – 3
2x = 8 - Isolate the variable “x”:
Divide both sides of the equation by 2 to isolate “x”:
2x / 2 = 8 / 2
x = 4
Therefore, the solution to the equation 2x + 3 = 11 is x = 4.
It is important to remember that the key to isolating variables in linear equations is to perform the same mathematical operations on both sides of the equation to maintain the equality. By following the steps outlined above and practicing with various examples, you can improve your skills in isolating variables in linear equations.
