91Ó°ÊÓ

When solving linear equations, the main goal is to find the value of the variable. This process is called 'isolating the variable.' To do this, you need to rearrange the equation so that the variable you are solving for is on one side of the equation and all other terms are on the opposite side. For example, consider this equation: \(8x - 15 = 7x + 3\). Here, you first move the terms involving the variable to one side and the constant terms to the other side. This helps to simplify the equation and isolate the variable. In step 1 of the solution given, subtraction of \(7x\) from both sides results in: \(8x - 7x - 15 = 7x - 7x + 3\). Simplifying this gives us: \(x - 15 = 3\). This leaves us with the variable \(x\) isolated on one side of the equation.

Sometimes, you will encounter equations that have variables on both sides. These are called 'equations with variables on both sides.' To solve these equations, you first need to move all the variable terms to one side of the equation. For example, in the exercise's equation \(8x - 15 = 7x + 3\): You have the \(x\) variable on both sides of the equation. To proceed, you subtract \(7x\) from both sides to get all the \(x\) terms on one side: \(8x - 7x - 15 = 7x - 7x + 3\). This simplifies to: \(x - 15 = 3\). By performing this step, we effectively make the equation simpler. From there, we can solve for the single variable.

Balancing equations is fundamental to solving any equation. The basic premise is that whatever operation you perform on one side of the equation, you must also perform on the other side. This keeps the equation balanced. In the given problem, these steps are clear: First, balance the variable terms: \(8x - 7x = 7x - 7x +3 \). This operation leaves us with \(x - 15 = 3\). To balance the equation further and isolate the variable, add 15 to both sides: \(x - 15 + 15 = 3 + 15\). This simplifies to: \(x = 18\). Now, the equation is balanced, and we have found the value of the variable.