91Ó°ÊÓ

Simplifying equations is an essential step when solving linear equations. It involves reducing the equation to its simplest form, making it easier to solve. In our example, we start with the equation \(x + 5x - 5 = 1\).

To simplify this equation, we first need to focus on reducing the number of terms. This means looking for similar items that can be merged together to make the equation less complicated. In our case, the terms \(x\) and \(5x\) can be combined, as they both involve the variable \(x\).

• Combine the terms \(x\) and \(5x\) on the left side of the equation.

When you add \(x + 5x\), you get \(6x\), so the equation simplifies to \(6x - 5 = 1\).

Simplification helps make the equation easier to manipulate and solve, setting the stage for the next steps.

Combining like terms is a vital process where you bring together terms that have the same variable component. This helps in reducing the equation to a more straightforward form. In our problem, we dealt with the equation \(x + 5x - 5 = 1\).

Look at the terms on the left side of the equation: \(x\) and \(5x\). Both are considered 'like terms' because they share the same variable component, \(x\).

• 'Like terms' are terms that have exactly the same variable parts. In this case, both terms have the variable \(x\).
• Adding them together gives us \(6x\).

This means the original equation can be rewritten as \(6x - 5 = 1\).

By combining these like terms, you've condensed the equation down to a more manageable size, making it easier to focus on the steps that follow, like isolating the variable.

Isolating the variable is the final major step in solving a linear equation. This involves manipulating the equation so that the variable is by itself on one side of the equation, allowing for its value to be found readily. Our example provides a clear demonstration of this process.

After simplifying and combining like terms, the equation is \(6x - 5 = 1\). To isolate \(x\), you start by moving the constant term to the other side. Add 5 to both sides, resulting in \(6x = 6\).

• Addition or subtraction can be used to move constants (numbers without variables) across the equals sign.

Next, divide each side by 6 to solve for \(x\). This is done because \(x\) is currently multiplied by 6. By dividing both sides by 6, you neutralize the multiplication.

• Division or multiplication can be employed to adjust the coefficient of the variable to 1.

The equation now reads \(x = \frac{6}{6}\), which simplifies to \(x = 1\).

Isolating variables facilitates finding the solution to the equation, as the final value of the variable is clearly presented.