If \(a / b=-3\) and \(a b=-12\), then: A. \(a-b=8\) B. \(a-b=-8\) C. \(a-b=8\) or -8 D. \(a-b=9\) E. \(a-b=4\) or -4

Algebraic equations involve finding the unknown values that make two expressions equal. In this exercise, we started with two equations: \[ \frac{a}{b} = -3 \] and \[ a \times b = -12 \]. To solve these, we need to express one variable in terms of another. We used the first equation to express \(a\) in terms of \(b\). The process included: \[ a = -3b \] This substitution simplifies our problem by reducing it to one variable. Then, we substituted this into the second equation to simplify further and solve for \(b\). Algebraic manipulation involves these key steps: expressing variables, substituting values, and simplifying expressions to find solutions.

The substitution method helps solve systems of equations by replacing one variable with an expression involving another. Here’s how it works:
1. Solve one of the equations for one variable in terms of the other: Since \[ \frac{a}{b} = -3 \], we found \( a = -3b \). 2. Substitute this expression into the other equation: Replace \(a\) in \( ab = -12 \) with \( -3b\), giving us: \[ (-3b)b = -12 \]. 3. Simplify and solve for the remaining variable: We simplified this to \[ -3b^2 = -12 \], which then gave \[ b^2 = 4 \]. Solving for \( b \), we found \[ b = \pm 2 \]. Finally, substituting these values back to find \( a \) gave us a full solution. The power of substitution lies in its ability to transform complex problems into simpler steps. It's a fundamental skill in algebra.

When solving algebraic equations, especially quadratic ones, you'll encounter both positive and negative solutions. For \( b^2 = 4 \), we found: \[ b = 2 \quad \text{or} \quad b = -2 \]. Each solution must be tested to determine corresponding values. For \( b = 2 \): \[ a = -3(2) = -6 \] For \( b = -2 \): \[ a = -3(-2) = 6 \] These yielded two computations for \( a - b \):
- For \( a = -6 \) and \( b = 2 \), \[ a - b = -8 \] - For \( a = 6 \) and \( b = -2 \), \[ a - b = 8 \] Therefore, the solutions for \( a - b \) are \[ 8 \quad \text{or} \quad -8 \], showing that quadratic equations often produce both positive and negative answers. Understanding this helps in solving more complex algebraic problems confidently.