Question

In each of the following, find (if possible) conditions on \(a\) and \(b\) such that the system has no solution, one solution, and infinitely many solutions. a. \(\begin{aligned} & x-2 y=1 \\ & a x+b y=5 \end{aligned}\) \(\begin{aligned} \text { b. } & x+b y=-1 \\ & a x+2 y=5 \end{aligned}\) c. \(\quad x-b y=-1\) \(x+a y=3\) d. \(\quad a x+y=1\) \(2 x+y=b\)

Short answer

Check conditions for intersection: parallel, intersect, or coincident for each pair of equations.

Step-by-step solution

Analyze System a

For the first system, we have two equations: \( x - 2y = 1 \) and \( ax + by = 5 \). To find conditions on \( a \) and \( b \):- **No Solution:** Means the lines are parallel. So, \( \frac{1}{a} = \frac{-2}{b} \) and \( \frac{1}{a} eq \frac{5}{1} \). Simplifying gives \( 2a = -b \) and \( 5a eq 1 \).- **One Solution:** Lines intersect, implying \( \frac{1}{a} eq \frac{-2}{b} \).- **Infinitely Many Solutions:** Lines are coincident, meaning \( \frac{1}{a} = \frac{-2}{b} = \frac{5}{1} \), which implies \( a = \frac{1}{5} \) and \( b = -\frac{2}{5} \).

Analyze System b

Now consider the system \( x + by = -1 \) and \( ax + 2y = 5 \).- **No Solution:** Lines are parallel, which gives \( \frac{1}{a} = \frac{b}{2} \) and \( \frac{-1}{a} eq \frac{5}{a} \). Simplifying, \( b = 2a \) and \( 5a eq -1 \).- **One Solution:** Lines intersect, \( \frac{1}{a} eq \frac{b}{2} \).- **Infinitely Many Solutions:** Same line, so \( \frac{1}{a} = \frac{b}{2} = \frac{-1}{5} \), leading to \( a = -5 \) and \( b = -10 \).

Analyze System c

For the system \( x - by = -1 \) and \( x + ay = 3 \):- **No Solution:** System of parallel lines, gives \( 1 = 1 \) and \( b = -a \) with \(-1 eq 3\), which is a contradiction because \( a eq -b \) always holds.- **One Solution:** Lines intersect, meaning \( b eq -a \).- **Infinitely Many Solutions:** Coincident lines, impossible here since \(-1 eq 3 \). Thus, this condition can't be satisfied.

Analyze System d

Next, \( ax + y = 1 \) and \( 2x + y = b \):- **No Solution:** Parallel lines require \( a = 2 \) while \( 1 eq b \), giving \( a = 2 \), \( b eq 1 \).- **One Solution:** Lines intersect, requiring \( a eq 2 \).- **Infinitely Many Solutions:** Coincident lines, so \( a = 2 \) and \( b = 1 \).

Key concepts

Parallel and Coincident Lines

When dealing with linear systems in algebra, encountering parallel and coincident lines is a common occurrence. Understanding these types of lines helps to determine the nature of solutions a system of equations might have.

• **Parallel Lines** arise when two lines have the same slope but different y-intercepts, meaning they never meet. In linear equations, this translates to having equal ratios of the coefficients of the variables, but unequal constant terms. For example, if the coefficients of the equations are such that \( \frac{1}{a} = \frac{-2}{b} \) and the constants differ, this indicates parallelism, leading to no solutions.


• **Coincident Lines** are essentially the same line and thus have the same slope and intercept. In algebraic terms, the equations of two lines are scalar multiples of each other, implying infinity of common points. This occurs when all coefficients and constants maintain a constant ratio. For example, \( \frac{1}{a} = \frac{-2}{b} = \frac{5}{1} \) signifies that both lines overlap perfectly, resulting in infinitely many solutions.

Linear Equations

Linear equations form the backbone of understanding systems of equations. They represent straight lines when plotted on a graph and are of the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. An equation such as \( ax + by = c \) rearranges to this format by solving for \( y \).

• **Slope:** It describes how steep a line is. In a linear equation \( y = mx + c \), \( m \) defines this slope. For example, in the equation \( x - 2y = 1 \), the slope is \( \frac{1}{2} \).


• **Y-intercept:** This tells where the line crosses the y-axis. For \( x - 2y = 1 \), rearrange it to \( y = \frac{1}{2}x - \frac{1}{2} \), revealing a y-intercept of \( -\frac{1}{2} \).

Understanding the structure of linear equations allows you to gauge if two lines are parallel, coincident, or intersecting by examining the slope and intercept thoroughly.

Intersection of Lines

The intersection of lines in a linear system determines whether there is a unique solution or not. When two lines intersect, they meet at a single point, indicating exactly one solution.

• **Unique Solution:** This occurs when the slopes of the two lines differ, implying the lines cross each other at a single point. Mathematically, this is represented by unequal ratios of the coefficients, like \( \frac{1}{a} eq \frac{-2}{b} \), ensuring that the lines are not parallel or coincident.


• An 'Intersection' describes these points where two lines meet. Solving for this intersection involves finding a solution that satisfies both equations simultaneously, providing the unique pair \((x, y)\).

Thus, knowing how to analyze intersections is vital for solving linear systems effectively, ensuring whether they yield one solution, none, or infinitely many.