Problem 101 - Algebra and Trigonometry

Question

The equation $$3 x+k-5=k x-k+1$$ is really a family of equations, because for each value of \(k\), we get a different equation with the unknown \(x\). The letter \(k\) is called a parameter for this family. What value should we pick for \(k\) to make the given value of \(x\) a solution of the resulting equation? (a) \(x=0 \quad\) (b) \(x=1 \quad\) (c) \(x=2\)

Step-by-Step Solution

Answer

For (a) \(k = 3\), for (b) \(k = 1\), and for (c) \(k = \frac{7}{5}\).

Step 1: Simplify the Equation

Begin by simplifying both sides of the equation: \(3x + k - 5 = kx - k + 1\). Distribute and combine like terms on both sides.

Step 2: Rearrange Equation for Given x Values

Plug in each value of \(x\) provided in the choices (\(x = 0, 1, 2\)) into the simplified equation to form equations in terms of \(k\).

Step 3: Solve for k for each x Value

Solve for \(k\) for each case:1. For \(x = 0\), simplify \(3(0) + k - 5 = k(0) - k + 1\) and solve.2. For \(x = 1\), simplify \(3(1) + k - 5 = k(1) - k + 1\) and solve.3. For \(x = 2\), simplify \(3(2) + k - 5 = k(2) - k + 1\) and solve.

Step 4: Check Each Value of k

Verify each solution by substituting the value of \(k\) back into the original equation to ensure consistency across both sides of the equation for the given \(x\).

Key Concepts

Parameters

In mathematics, a parameter is a special variable that is used to define a family of equations. It works as a fixed number that can change, creating different versions of the equation without altering its basic structure. In the exercise we're looking at, the variable \( k \) is the parameter.
- **Parameter Role**: Here, \( k \) helps form different equations for each value we assign to it.- **Effect of Changing Parameter**: By changing \( k \), the equation shifts, allowing us to explore how solutions for \( x \) vary.
Knowing how parameters work can help in understanding models in different fields, such as physics and economics, where parameters define scenarios or conditions.

Equation Simplification

Simplifying an equation involves reducing it to its most basic form. This process makes solving the equation easier and clearer. We often focus on combining like terms and performing arithmetic operations while simplifying.
- **Identify and Combine**: First, look for like terms on both sides of the equation. For example, in the problem's equation, we find terms involving \( x \) and constants that can be grouped together.- **Rearrange Terms**: Carefully arrange terms to make the equation more straightforward. The initial equation \(3x + k - 5 = kx - k + 1\) becomes simpler once you organize and regroup these parts.Simplification is like tidying up a room, making everything cleaner and more ordered, laying the groundwork for finding a solution.

Solving Equations

Solving equations is about finding the value for an unknown variable—here it's \( k \) when \( x \) is specified. We often substitute known values and simplify the resulting equation to reveal the solution.
- **Substitution**: Replace \( x \) with given numerical values to work out distinct equations for each case.

• For \( x = 0 \), plug it into the equation to form a simple linear equation in terms of \( k \).
• Repeat this process for \( x = 1 \) and \( x = 2 \).

- **Solve for Parameter**: Each time, isolating \( k \) on one side of the equation to see its possible values. This involves basic algebraic manipulation, such as adding or subtracting terms.Being able to solve for a parameter by substituting different values provides insight into how parameters affect solutions. This process is crucial in many areas, from determining interest rates in finance to calculating forces in engineering.