91Ó°ÊÓ

When solving linear equations, one of the essential tools at your disposal is the distributive property. This property states that for any real numbers a, b, and c, the equation \(a(b + c) = ab + ac\) holds true. In essence, it allows you to remove parentheses by distributing the multiplication over addition or subtraction inside the brackets.

For example, if we look at the original problem \(5x - (2x - 10) = 35\), we apply the distributive property by multiplying -1 (the negative sign can be seen as -1) by each of the terms inside the parentheses. This transforms \(2x - 10\) into \(-2x + 10\), effectively distributing the negative sign. Understanding this concept is critical for simplifying equations and setting the stage for further steps in solving.

The next step in solving equations after applying the distributive property often involves combining like terms. 'Like terms' are terms that contain the same variables raised to the same power and can therefore be combined. To combine them, simply add or subtract their coefficients.

In our example, after distributing the negative sign, we obtain the equation \(5x - 2x + 10 = 35\). Here, \(5x\) and \(-2x\) are like terms. We combine them by subtracting their coefficients to get \(3x\), leading to a simplified equation of \(3x + 10 = 35\). It's important to carefully identify and combine like terms correctly to accurately simplify the equation.

The final thrust in solving an equation is to isolate the variable. This involves manipulating the equation such that the variable stands alone on one side, clearly displaying the solution. We achieve this by performing operations that 'undo' what is being done to the variable, like adding, subtracting, multiplying, or dividing.

In the exercise, after combining like terms, we obtained \(3x + 10 = 35\). To isolate x, subtract 10 from both sides of the equation, giving us \(3x = 25\). The last step is to divide each side by 3 to leave the variable x by itself, ultimately finding that \(x = \frac{25}{3}\) or approximately \(x ≈ 8.33\). By isolating the variable, we directly find the solution to the equation, completing the process.