91Ó°ÊÓ

In algebra, the concept of like terms is a foundational building block for combining terms successfully in an equation or expression. Like terms are those terms in an expression that have the same variable raised to the same power. For example, in the original exercise, the terms

• \(-2.2x\)
• \(-7.8x\)

are considered like terms because they both involve the variable \(x\) raised to the same power of 1.

Combining like terms means adding or subtracting their coefficients, the numeric factors, while maintaining the common variable and exponent. In the example, we added \(-2.2\) and \(-7.8\) to get \(-10x\)\: \-2.2x - 7.8x = -10x\.
By doing this, we simplify the equation by reducing the number of terms, making it more manageable and easier to solve later on.

Variable isolation is a key step in solving any equation, as it aims to separate the variable on one side of the equation. This allows us to find its value by performing operations that simplify the equation sequentially.

In the original problem, once like terms were combined, the equation was reduced to:

• \(-10x - 0.8 = -3.3\)

To isolate the \(x\)-term, you need to eliminate the constant term on its side, which is \-0.8\. This is done by performing the opposite operation, in this case, adding 0.8 to both sides:\[-10x - 0.8 + 0.8 = -3.3 + 0.8\],This results in\[-10x = -2.5\]By isolating \(x\), we make it the only variable in the equation, ready for the final steps to find its specific numerical value.

Division in equations is frequently the last step in isolating the variable completely. It is used to solve for the variable once it is alone on one side of the equation but still multiplied by a number.

In our example, the simplified equation after combining like terms and isolating \(x\) was:\[-10x = -2.5\]
This equation indicates that \(x\) is multiplied by \-10\ and needs to be divided by the same coefficient to get \(x\) by itself. So, we perform the operation:\[x = \frac{-2.5}{-10}\],which simplifies through division, resulting in:\[x = 0.25\]
Keep in mind that dividing both sides by the same number because it maintains the balance of the equation, preserving its equality. Understanding and applying division appropriately propels us to find the solution for \(x\). This underlying concept is essential in achieving the correct final answer in math problems involving equations.