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Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes an equation true. One standard type of equation is the linear equation. A linear equation can be written in the form: \(ax + b = c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we need to solve for. To solve, follow clear steps to manipulate the equation.
In our example, consider the equation: \(4x + 3 = 27\). This equation is linear because \(x\) appears only to the first power. The goal is to figure out what \(x\) equals when this equation holds true.

The process involves a couple of straightforward steps. First, isolate the variable term on one side of the equation. Then, perform any necessary arithmetic to solve for the variable.

Isolating the variable is a crucial part of solving linear equations. The goal is to get the variable \(x\) by itself on one side of the equation. This makes it easy to see what \(x\) equals.
In the example, we start with the equation: \(4x + 3 = 27\). To isolate \(x\), we first need to get rid of the constant term on the left side.

Here’s how it’s done:

• Subtract 3 from both sides of the equation:
• \(4x + 3 - 3 = 27 - 3\)
• This simplifies to: \(4x = 24\)

Now that we have \(4x\) alone, we need to solve for \(x\). Divide both sides of the equation by 4:

• \(x = \frac{24}{4}\)
• This simplifies to: \(x = 6\)

By isolating the variable, we’ve found that \(x\) equals 6.

Once you've isolated the variable and solved the equation, it's important to check your solution. This step ensures your answer is correct and your equation manipulation was done accurately.
Let’s check the solution of our example where we found \(x = 6\). To verify this, substitute \(x = 6\) back into the original equation: \(4x + 3 = 27\).

Substituting 6 for x, we get:

• \(4(6) + 3 = 27\)
• Simplify the left side: \(24 + 3 = 27\)
• It is correct because \(27 = 27\)

This confirms that the solution \(x = 6\) is indeed correct.

Always remember, checking your solution is like double-checking your work. It ensures accuracy and reinforces your understanding of the problem-solving process.