Problem 80 - Precalculus

Question

For the system of equations $$ \begin{array}{l} 3 x+2 y=5 \\ a x-4 y=1 \end{array} $$ find \(a\) that guarantees no unique solution.

Step-by-Step Solution

Answer

The value of \(a\) is \(-6\).

Step 1: Identify the Condition for No Unique Solution

For a system of equations to have no unique solution, the two equations must be linearly dependent. This means that one equation is a scalar multiple of the other. Therefore, the ratios of the coefficients of the variables and the constants must be equal.

Step 2: Set Up the Ratios

For the system of equations \(3x + 2y = 5\) and \(ax - 4y = 1\), set up the ratios for the coefficients of \(x\), \(y\), and the constants:\[ \frac{3}{a} = \frac{2}{-4} = \frac{5}{1} \]

Step 3: Solve the Ratio for \(a\)

Since the second equation \( \frac{2}{-4} \) simplifies to \( -\frac{1}{2} \), equate \( \frac{3}{a} = -\frac{1}{2} \):\[ 3 = -\frac{a}{2} \]Multiply both sides by \(-2\) to solve for \(a\):\[ a = -6 \]

Step 4: Verify the Ratios

Verify the value of \(a\) by checking the constant ratio:Even though \( \frac{5}{1} eq -\frac{1}{2} \), the ratio \( -\frac{1}{2} \) from the coefficients confirms consistent dependence without a solution, as it indicates either the system has no solution or infinitely many solutions, depending on the constants.

Key Concepts

Linear Dependency

In mathematics, especially when dealing with systems of equations, the concept of **linear dependency** comes into play. Linear dependency arises when one equation in the system can be derived from another by simply multiplying each term by the same number, also known as a scalar. When two equations are linearly dependent, they do not provide unique intersecting points as solutions, but rather describe the same line or lie on parallel lines.
For example, consider the following simple case: if you have an equation \(3x + 2y = 5\) and wish to make another equation \(ax - 4y = 1\) linearly dependent, it implies equation-related transformations that maintain proportionality across all parts of the equation.

Scalar Multiple

A **scalar multiple** is a concept where you multiply every term of an equation by a constant value, known as a scalar. This scalar keeps the equation equivalent in terms of linear combinations or dependencies but changes its appearance.
For example, if you have two equations such as \(3x + 2y = 5\) and another equation like \(ax - 4y = 1\), one can explore values of \(a\) by expressing each equation as a multiple of the other. This is done by finding a common ratio where each term, including the constant term, complies with this multiplication factor.

Coefficients

**Coefficients** are the numerical factors that multiply variables in equations. In systems of linear equations, examining the coefficients is crucial in determining the relationship between the equations.

• In the equation \(3x + 2y = 5\), the coefficients are \(3\) for \(x\) and \(2\) for \(y\).
• In the second equation, \(ax - 4y = 1\), the coefficients are \(a\) for \(x\) and \(-4\) for \(y\).

When looking for no unique solution, the coefficients must linearly relate. Analyzing these numbers helps to spot scalar multiples or to assert if the equations are linearly dependent.

No Unique Solution

1. A system has **no unique solution** when the equations are either identical or parallel, meaning they do not intersect at exactly one point. This occurs when the equations are linearly dependent by the previous concepts of scalar multiples and coefficients.
Consider the system:
\[3x + 2y = 5\]
\[ax - 4y = 1\] When transformed appropriately utilizing the ratios between coefficients, if \(a = -6\), the ratios, alongside the constants, determine the type of solutions available.

• If all ratios match and constants align, the system has infinitely many solutions.
• If coefficients align via a scalar, but the constant doesn't match, then there are no intersections, resulting in no solutions.

This understanding leverages these key ideas and transforms coefficients to determine how systems resolve.