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When solving linear equations, a crucial first step is isolating the variable you want to solve for. This means you want to get the variable on one side of the equation and everything else on the other side. For example, in the equation \(2b + 7 = b + 10\), the goal is to get \(b\) by itself. Subtract \(b\) from both sides to do this: \(2b + 7 - b = b + 10 - b\), which simplifies to \(b + 7 = 10\). This isolation helps you to see clearly what the value of the variable should be. Isolating variables often involves

• adding or subtracting terms on both sides of the equation,
• and sometimes multiplying or dividing both sides to remove coefficients.

Once the variable is isolated, you can move on to the next step: solving for the variable.

After isolating the variable, the next step is to solve for it, i.e., to find its value. Continuing from our previous example, we have the equation: \(b + 7 = 10\). To solve for \(b\), subtract 7 from both sides: \(b + 7 - 7 = 10 - 7\), simplifying to \(b = 3\). In another example, let's consider the equation \(3b - 4 = 24 - b\). Start by isolating \(b\) on one side. Add \(b\) to both sides so you have \(3b - 4 + b = 24 - b + b\). This simplifies to \(4b - 4 = 24\). Now, solve for \(b\) by adding 4 to both sides: \(4b - 4 + 4 = 24 + 4\). This yields \(4b = 28\). Finally, divide both sides by 4 to isolate \(b\): \(b = \frac{28}{4}\), which simplifies to \(b = 7\). Each of these actions—adding, subtracting, multiplying, or dividing—is used to isolate and solve for the variable.

Understanding how basic algebra works is fundamental to solving equations. At its core, algebra is about finding unknown values. Here are some important concepts:

• Variables: Symbols like \(b\) that represent unknown numbers.
• Equations: Mathematical statements that assert equality between two expressions, like \(2b + 7 = b + 10\).
• Operations: Addition, subtraction, multiplication, and division used to manipulate equations and solve for variables.

For example, in the equation \(3b - 4 = 24 - b\), you use addition to get all \(b\) terms on one side and constants on the other. Then, you solve for the variable using basic arithmetic operations. Knowing these basic algebraic operations makes it easier to handle more refined and complex problems as you advance in mathematics. Practice frequently to get a good grasp of these concepts. That’s the foundation for becoming proficient in algebra!