91Ó°ÊÓ

When facing a linear equation, the first step often involves simplifying each side by combining like terms. Like terms are parts of an equation that have the same variable raised to the same power. In our exercise,

• We have constants on the left side of the equation: \(-2\) and \(-10\).
• Since both expressions are constants (not involving any variable), we can add them together.

By doing this, we simplify the expression \(-2 - 10\) to \(-12\).
Combining like terms helps to simplify equations and make it easier to solve for the variable. It’s important to always look out for terms that can be combined on each side of an equation. This makes the task of solving the linear equation more straightforward.

In many equations, you might find fractions, which can complicate solving the equation. One efficient way to handle fractions is to eliminate them by multiplying both sides of the equation by a common denominator. In our example, what we have is:

• The term \(\frac{z}{10}\), which is a fraction.

To eliminate this fraction, we multiplied both sides of the equation by \(10\), which is the denominator of the fraction. This gives us \(10 \times (-12) = 10 \times \frac{z}{10}\).
Multiplying by the common denominator cancels the fraction on the side containing the variable and simplifies the equation. It is a crucial step that helps transition from a fraction form to a more manageable linear equation, making it easier to isolate the variable.

The ultimate goal when solving a linear equation is to find the value of the variable. This means isolating the variable on one side of the equation.

• After eliminating the fraction in our equation, we arrive at \(-120 = z\).

Here, the variable \(z\) is already isolated on one side of the equation.
Isolating the variable is often the final step, but it is a vital one. It gives us a clear view of what the solution to the equation is. This method reflects the core idea of solving equations: to rewrite the equation in a way that shows exactly what the value of the unknown variable is. Always ensure that the variable is on one side by itself for accuracy and clarity in the result.