Problem 36 - Thomas Calculus

Question

Is there anything special about the relationship between the lines \(A x+B y=C_{1}\) and \(A x+B y=C_{2}(A \neq 0, B \neq 0) ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

The lines are parallel because they share the same slope but have different intercepts.

Step 1: Identify the Lines

First, let's look at the two given equations: 1. Line 1: \( A x + B y = C_1 \)2. Line 2: \( A x + B y = C_2 \).These are equations of two straight lines.

Step 2: Standard Form Observation

Notice that both equations have the same coefficients for \( x \) and \( y \). This indicates that the lines have the same slope.

Step 3: Determine Parallelism

Since the coefficients of \( x \) and \( y \) are identical in both lines, this implies that their slope is the same. Therefore, the lines are parallel to each other.

Step 4: Parallel Lines and Intercepts

Parallel lines never intersect. The difference between \( C_1 \) and \( C_2 \) represents the distance between the two parallel lines on the graph.

Step 5: Conclude the Relationship

Given that \( A x + B y = C_1 \) and \( A x + B y = C_2 \) have the same slope but different intercepts, they are parallel lines with no points of intersection.

Step 6: Reasoning

The primary reason the lines are parallel is that they differ solely in their constants \( C_1 \) and \( C_2 \), while maintaining the same \( x \) and \( y \) coefficients, indicating consistent slope.

Key Concepts

Straight Line Equations

A straight line equation can be represented in its standard form as \( Ax + By = C \). This form highlights the coefficients of the variables \( x \) and \( y \). These coefficients are essential because they help determine the slope of the line.
The "C" in the equation represents a constant that shifts the line up or down vertically on a graph, depending on its value. Changing "C" affects where the line intersects the y-axis but does not change its direction.

• **Standard Form**: \( Ax + By = C \)
• **Factors**: \( A \), \( B \) (coefficients of \( x \) and \( y \))
• **Constant**: "C", which affects the line's position

Understanding the structure of this equation is essential when analyzing multiple lines, especially to determine if they're parallel or not.

Slope

The concept of slope is pivotal when discussing straight lines. The slope refers to the line's steepness or its angle of incline. In the standard line equation, the slope "m" can be calculated using the formula \( m = -\frac{A}{B} \).
If lines have identical slopes, they are always parallel. This is because they tilt at the same angle and therefore can never meet.
When examining our equations \( Ax + By = C_1 \) and \( Ax + By = C_2 \), you will notice that both have the same coefficients for \( x \) and \( y \). This results in equal slopes.

• **Slope Formula**: \( m = -\frac{A}{B} \)
• **Identical Slopes**: Means lines are parallel
• **Significance**: Determines whether lines intersect or run parallel

Recognizing that slopes are identical is key to identifying parallel lines.

Intercepts

Intercepts give us information about where a line crosses the axes on a graph. There are two types: x-intercept and y-intercept. In the context of our straight line equations, each change in constant "C" will alter the intercepts without affecting the slope.

x-Intercept

The x-intercept occurs where the line crosses the x-axis, i.e., \( y = 0 \).
For the equation \( Ax + By = C \), we find: \[ x = \frac{C}{A} \].

y-Intercept

The y-intercept is where the line crosses the y-axis, i.e., \( x = 0 \).
For the same equation, it becomes: \[ y = \frac{C}{B} \].

• **x-Intercept**: \( x = \frac{C}{A} \)
• **y-Intercept**: \( y = \frac{C}{B} \)
• **Interception Change**: Different "C" values shift the intercepts

Understanding intercepts helps us see how lines can shift up or down without changing direction.

Graphical Representation

Graphical representation is an effective way to visualize the relationships between lines, such as our original problem with equations \( Ax + By = C_1 \) and \( Ax + By = C_2 \). Lines with identical slopes will appear as parallel on a graph, consistently distant from one another. This gap is dictated by the difference between their constants \( C_1 \) and \( C_2 \).

Visualizing Parallel Lines

When graphed, parallel lines will never cross. Their consistent slope visually affirms their parallelism. As \( C_1 \) differs from \( C_2 \), it simply relocates the line vertically.

• **Parallel Visualization**: Equidistant lines on a graph
• **Constant Difference**: Reflects vertical displacement
• **Graph Reading**: Aids in understanding line position and movement

A graph clearly shows that even as intercepts change or lines move, parallel lines remain equidistant and never overlap.