Problem 24 - Algebra Form and Function

Question

Without solving them, say whether the equations have a positive solution, a negative solution, a zero solution, or no solution. Give a reason for your answer. $$ 6 p=9 p-4 $$

Step-by-Step Solution

Answer

Answer: The linear equation has a positive solution.

Step 1: Analyze the equation

After analyzing the given linear equation, we have: $$ 6p = 9p - 4 $$ We can notice that the equation involves only one variable, \(p\), which can take on any real value. This means there's no immediate restriction on the possible values of \(p\). Next, let's work on the equation to find any constraints that might apply to this variable.

Step 2: Check for constraints

To identify any constraints that would affect the possible solutions for \(p\), let's rearrange the equation by moving all terms to one side: $$ 9p - 6p = 4 $$ Now, simplify the equation: $$ 3p = 4 $$ Since all the terms are now on one side and the equation has been simplified, we can proceed to analyze the possible solutions for \(p\).

Step 3: Determine the type of solution

We have got the simplified equation: $$ 3p = 4 $$ To find the type of solution without actually solving the equation, we can consider the signs of the coefficients and the constant term. - The coefficient of \(p\) is \(3\) which is a positive number. - The constant term \(4\) is also a positive number. From this information, we can deduce that the solution to this equation must be a positive number. This is because dividing a positive number by a positive number will result in a positive number. Thus, the equation has a positive solution.

Key Concepts

Solution Types in Linear Equations

When dealing with linear equations, understanding the types of solutions you might encounter is important. Linear equations typically have one of three types of solutions:

• Positive Solution: Here, the result of solving the equation gives a positive value for the variable.
• Negative Solution: This occurs when solving the equation results in a negative value.
• Zero Solution: When the equation is structured such that the variable equals zero.
• No Solution: Sometimes, rearranging and simplifying an equation may reveal no valid solutions exist.

In the given exercise, after simplifying to equation \(3p = 4\), both terms are positive suggesting a positive solution. Thus, when the solution leads to a positive outcome, as in this case, the solution type is said to be positive.

Variable Constraints

Constraints on variables in equations dictate what values the variable can take. Often, these constraints arise from the context of the problem or inherent properties of the equation itself.
Some constraints to consider include:

• Domain restrictions: In a real-world context, a variable might need to be non-negative if it represents something like a length or amount of items.
• Mathematical operations: The equation \(\frac{1}{x}\) implies \(x\) cannot be zero as division by zero is undefined.
• Simplification and rearrangement: Moving terms around, as in the original exercise from \(6p = 9p - 4\) to \(9p - 6p = 4\), helps clarify potential constraints.

For the exercise provided, rearranging yields no restrictive constraints on \(p\). Therefore, the solution can be any real number that satisfies the equation once reduced to \(3p = 4\).

Equation Analysis

Equation analysis is the process of breaking down and understanding equations to determine their solvability and the nature of their solutions. This involves several steps:
1. Variable Isolation: Identify the variable of interest, here it's \(p\), and rearrange the equation so that this variable stands alone on one side if possible.
2. Simplification: Combine like terms and reduce the equation, as in \(6p = 9p - 4\) simplifying to \(3p = 4\) by subtracting \(6p\) from both sides.
3. Sign Analysis: For the given equation, note the positive coefficient \(3\) of \(p\) and the positive constant 4. This simplifies decision-making concerning solution type since dividing a positive constant by another positive number yields a positive solution.
Equation analysis assists in understanding the nature of the solutions and any constraints, enabling a solution that is logical and valid. In this case, the analysis shows a straightforward linear relationship leading to a positive solution.