Problem 91 - Algebra for College Students

Question

Consider the linear system \(\left(\begin{array}{l}a_{1} x+b_{1} y=c_{1} \\\ a_{2} x+b_{2} y=c_{2}\end{array}\right)\). (a) Prove that this system has exactly one solution if and only if \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\). (b) Prove that this system has no solution if and only if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} .\) (c) Prove that this system has infinitely many solutions if and only if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\).

Step-by-Step Solution

Answer

(a) One solution: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \). (b) No solution: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \). (c) Infinite solutions: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).

Step 1: Understanding the System of Equations

The given system of equations is: \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \). The solution to this depends on the relationships among their coefficients.

Step 2: Part (a): One Solution Scenario

For the system to have exactly one solution, the lines represented by the equations must intersect. This occurs when the lines are not parallel, which is when their slopes are different. The slopes for the equations are \( -\frac{a_1}{b_1} \) and \( -\frac{a_2}{b_2} \). The condition for different slopes is \( \frac{a_1}{a_2} eq \frac{b_1}{b_2} \). Thus, the system has exactly one solution if and only if \( \frac{a_1}{a_2} eq \frac{b_1}{b_2} \).

Step 3: Part (b): No Solution Scenario

The system has no solution if the lines are parallel but distinct, which means they have the same slope but different intercepts. This occurs when \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) (same slope) but \( \frac{c_1}{c_2} \) is different (different intercept). Therefore, the system has no solution if and only if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} eq \frac{c_1}{c_2} \).

Step 4: Part (c): Infinitely Many Solutions Scenario

For the system to have infinitely many solutions, the lines must coincide, meaning they represent the same line. This happens when all corresponding coefficients are in the same ratio: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \). Thus, the system has infinitely many solutions if and only if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).

Key Concepts

Unique Solution Condition

A system of linear equations has a unique solution when the lines represented by these equations intersect at exactly one point. This means the slopes of the lines must be different, ensuring they eventually cross each other. In mathematical terms, the slope of a line from an equation of the form \( ax + by = c \) is \( -\frac{a}{b} \).

• If two lines have different slopes, they will intersect at one unique point.
• This intersection occurs only if the ratio of the coefficients of the terms \(x\) and \(y\) (i.e., \( \frac{a_1}{a_2} \)) is not the same as the ratio \( \frac{b_1}{b_2} \).

Thus, according to the given exercise, a system has a unique solution if \( \frac{a_1}{a_2} eq \frac{b_1}{b_2} \). This assures that the lines intersect, and there is one specific solution to the system.

Parallel Lines

Parallel lines are a key concept when examining system solutions, particularly for no-solution scenarios. Lines are parallel if they have the same slope but different intercepts, meaning they will never meet. For the system \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \), the slopes are \( -\frac{a_1}{b_1} \) and \( -\frac{a_2}{b_2} \).

• When \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), it means both lines have the same slope.
• If intercepts differ, expressed as \( \frac{c_1}{c_2} eq \frac{b_1}{b_2} \), the lines are distinct and will never intersect.

In summary, parallel but distinct lines lead to no solutions, so \( \frac{a_1}{a_2} = \frac{b_1}{b_2} eq \frac{c_1}{c_2} \) results in a system with no solution.

Infinite Solutions

Infinite solutions occur when two lines are identical, also known as coincident lines. This means every point on one line lies also on the other, leading to infinitely many solutions. For this to happen, all corresponding coefficients in the equations must have the same ratio. Let's break it down:

• The equations share the same slope \( \frac{a_1}{b_1} = \frac{a_2}{b_2} \).
• The y-intercepts must also be identical, which is captured by the condition \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).

Therefore, if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the system represents the same line and thus has infinite solutions, effectively overlapping entirely.