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Algebra is a fundamental branch of mathematics that helps us solve equations through symbols and letters that represent numbers.
It involves finding unknown values, called variables, and expressing relationships between quantities. In algebra, we often use letters like \( q \) or \( x \) to stand for these unknown values. In a linear equation, the highest power of the variable is one, which makes solving these equations straightforward.
For example, given an equation like \( 96q + 23 = -265 \), our goal is to determine the value of \( q \). Linear equations like this one are key components in algebra because they describe real-world relationships and can be applied to solve practical problems.

Isolating variables is about getting the variable you're solving for on its own on one side of the equation. This is crucial because it allows you to find the value of the unknown variable.
In the equation \( 96q + 23 = -265 \), we begin isolating \( q \) by eliminating any added or subtracted numbers from its side.
Here’s how:

• Firstly, subtract 23 from both sides to remove it from the left side. This changes the equation to \( 96q = -265 - 23 \).
• Simplify the right side: \( -265 - 23 = -288 \), so now the equation is \( 96q = -288 \).
• Next, divide each side by 96 to solve for \( q \). This keeps the equation balanced and isolates \( q \) as \( q = -288/96 \).

When the variable is isolated and solved, as in \( q = -3 \), you’ve found its value. Isolating variables is a step-by-step process that ensures you neatly work through an equation and find the answer.

Checking solutions means substituting the value you've found back into the original equation to ensure it satisfies the equation. This step confirms that you have the correct solution, preventing errors in your work.
After solving for \( q \) in earlier steps, substitute \( q = -3 \) back into the given equation \( 96q + 23 = -265 \):

• Replace \( q \) with \( -3 \): \( 96(-3) + 23 \).
• Calculate \( 96 imes (-3) = -288 \).
• Add \( 23 \) to \( -288 \): \( -288 + 23 = -265 \).

Since the left side equals the right side of the original equation, \( -265 = -265 \), it confirms that \( q = -3 \) is indeed the correct solution. Checking solutions is an essential final step in solving equations, as it validates that your work is correct and complete.