Question

Find all solutions to the general system \(a x=b\) of one equation in one variable (a) when \(a=0\) and \((b)\) when \(a \neq 0\).

Short answer

If \(a = 0\), solutions depend on \(b\):\ \(b=0\) (infinitely many solutions); \(b \neq 0\) (no solution). If \(a \neq 0\), solution is \(x = \frac{b}{a}\).

Step-by-step solution

Analyze the Case when a = 0

When the coefficient \(a\) in the equation \(a x = b\) is zero, the equation becomes \(0 \cdot x = b\). This simplifies to \(0 = b\). This is a key observation to determine whether a solution exists. - If \(b = 0\), then \(0 = 0\), which is true for any value of \(x\). Therefore, there are infinitely many solutions for \(x\).- If \(b eq 0\), then \(0 = b\) is false, meaning there is no solution for \(x\).

Analyze the Case when a ≠ 0

When the coefficient \(a\) is non-zero, the equation \(a x = b\) can be solved for \(x\). Divide both sides of the equation by \(a\): \[ x = \frac{b}{a} \]Since \(a eq 0\), this division is valid and there will be exactly one solution, which is \(x = \frac{b}{a}\).

Summarize Solutions for Both Cases

Summarizing from the analysis:- If \(a = 0\) and \(b = 0\), there are infinitely many solutions.- If \(a = 0\) and \(b eq 0\), there is no solution.- If \(a eq 0\), there is exactly one solution, \(x = \frac{b}{a}\).

Key concepts

Infinite Solutions

In the world of linear equations, the concept of infinite solutions arises when an equation holds true for an infinite set of values. This is often encountered in simple algebraic equations such as the one given when the coefficient of a variable equals zero. Consider the equation \(a x = b\) where \(a = 0\). It transforms into \(0 \cdot x = b\), effectively reducing to the statement \(0 = b\).
When \(b\) is also zero, this statement turns true because zero equals zero, regardless of the value of \(x\).
Therefore, every possible value for \(x\) will satisfy the equation, leading to infinitely many solutions. This is an example of dependent equations—the equation does not "depend" on finding a specific number to satisfy it. This is unique to linear equations when a variable's multiplier is zero, as it dissolves the specific need to resolve for \(x\). Remembering this scenario helps in recognizing patterns in solving broader algebraic systems.

Unique Solution

The term unique solution refers to the scenario when exactly one value satisfies the given equation. For the equation \(a x = b\), a unique solution arises when \(a eq 0\). In this case, you can manipulate the equation to isolate \(x\). Divide both sides by \(a\) to arrive at \(x = \frac{b}{a}\).
This mathematical transformation works solely because \(a\), the coefficient of \(x\), is non-zero which makes every operation valid and reversible.
The unique solution ensures that only one specific value of \(x\) will satisfy the equation, defining an independent and distinct solution. This is typical in solving linear equations where coefficients are pivotal in determining the solution's nature. The unique resolutions are critical in fields ranging from simple calculations to elaborate modeling in sciences, explaining why linear equations lay the backbone of numerous mathematical domains.

Algebraic Problem-Solving

Algebraic problem-solving involves applying strategies and mathematical rules to find the unknown in equations. The example of finding solutions to \(a x = b\) entrenches the essence of this process.

To successfully solve algebraic equations:

• Identify the variables involved.
• Analyze coefficients and constants, determining if special cases like zero apply.
• Employ operations like addition, subtraction, multiplication, or division to restructure the equation for the unknown variable.

In the given equation, decision nodes appear—\(a = 0\) vs \(a eq 0\)—which lead to different solution paths. When \(a = 0\), further evaluate \(b\) to check for infinite solutions; when \(a eq 0\), rearrange the formula to solve for \(x\).
This illustrates how algebra guides us through logical reasoning, shaping approaches to break down complex problems into simpler, manageable parts.

Algebra is thus a powerful tool, often preparing learners to tackle varied disciplines by fostering exact thinking and precise solution crafting.