Find the slope of the line whose equation is [tex]$p x-q y=r+1$[/tex].
Answer (1)
Rewrite the given equation p x − q y = r + 1 .
Isolate the y term: − q y = − p x + r + 1 .
Divide by − q to get the slope-intercept form: y = q p x − q r + 1 .
Identify the slope as the coefficient of x : q p .
Explanation
Understanding the Problem We are given the equation of a line as p x − q y = r + 1 , and we need to find its slope. To do this, we will rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Isolating the y term First, we isolate the y term in the given equation:
p x − q y = r + 1
Subtract p x from both sides:
− q y = − p x + r + 1
Solving for y Next, we divide both sides of the equation by − q to solve for y :
y = − q − p x + r + 1
y = − q − p x + − q r + 1
y = q p x − q r + 1
Identifying the Slope Now, the equation is in the slope-intercept form y = m x + b . Comparing this with our result, we can see that the slope m is q p and the y-intercept b is − q r + 1 .
Final Answer Therefore, the slope of the line is q p .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, in construction, the slope of a ramp determines its steepness, affecting accessibility. In economics, the slope of a supply or demand curve indicates how sensitive the quantity is to changes in price. By finding the slope, we can analyze and predict the behavior of linear relationships in various fields, enabling informed decision-making and efficient problem-solving. For instance, if p = 2 and q = 3 , the slope is 3 2 , meaning for every 3 units we move horizontally, we move 2 units vertically.
