Problem 17 - 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. $$ 3 x=5 $$

Step-by-Step Solution

Answer

Answer: The given equation $$3x = 5$$ has a positive solution.

Step 1: Analyze the equation

Here, we have a linear equation in the form of $$ax = b$$, with $$a = 3$$ and $$b = 5$$. This type of equation typically has a unique solution for $$x$$.

Step 2: Determine the solution type

In this case, we have a linear equation with a positive coefficient for $$x$$ and a positive constant term. The left side of the equation is $$3x$$, which becomes a larger positive number as $$x$$ increases. As $$x$$ gets smaller and approaches negative values, the left side value will also decrease. Since the right side of the equation is $$5$$, a positive number, it is possible for the left side to equal the right side by having a positive value of $$x$$. Therefore, the given equation $$3x = 5$$ has a positive solution.

Key Concepts

Positive Solution

A positive solution in the context of linear equations means that the variable, usually represented by \(x\), takes a positive value that satisfies the equation. Linear equations follow the general form \(ax = b\), where \(a\) and \(b\) are constants. The solution is positive if increasing \(x\) on the left side produces the positive constant on the right side.

Consider the equation \(3x = 5\). Here, both \(a = 3\) and \(b = 5\) are positive. When we solve \(3x = 5\), we divide both sides by 3 to isolate \(x\), resulting in \(x = \frac{5}{3}\). This value of \(x\) is positive, indicating a positive solution.

Key points to remember:

• If both \(a\) and \(b\) are positive in the equation \(ax = b\), the solution for \(x\) is positive.
• A positive solution satisfies the equation and maintains the equality.

Unique Solution

Linear equations generally have a unique solution unless there's an identical equation or contradiction. When you have a single linear equation with a single variable, like \(ax = b\), there is always one specific value of \(x\) that solves the equation if \(a eq 0\).

For the equation \(3x = 5\), the uniqueness is evident. The equation’s form ensures that only one specific value of \(x\) will satisfy it. By performing the algebraic operation of dividing both sides by 3, we find that \(x = \frac{5}{3}\) is the sole solution.

Key points to note:

• Linear equations of the form \(ax = b\) have one unique solution as long as \(a eq 0\).
• This uniqueness comes from the fact that linear equations graph as straight lines, and a line intersects a horizontal plane at only one point.

Linear Equations with Positive Coefficients

Linear equations with positive coefficients imply that the number in front of \(x\) (the coefficient) is greater than zero. This property affects how the solution of the equation behaves.

In our given equation \(3x = 5\), \(a = 3\) is a positive coefficient. When \(a\) is positive in \(ax = b\), it generally indicates the following:

• The equation has a positive slope when graphed. This means the line rises as you move from left to right.
• If \(a\) and \(b\) are both positive, the solution for \(x\) will also be positive, as the equation demands a positive \(x\) to make \(ax = b\) true.

Understanding the impact of positive coefficients aids in predicting the type of solution and graphing the equation. This helps anticipate whether solutions will be positive, negative, or zero, especially in more complex systems of equations.