Find the slope of the line parallel to the line [tex]$2 x-3 y=6$[/tex]
Answer (1)
Rewrite the given equation 2 x − 3 y = 6 in slope-intercept form.
Isolate y to get y = 3 2 x − 2 .
Identify the slope as the coefficient of x , which is 3 2 .
State that the slope of the parallel line is the same, so the final answer is 3 2 .
Explanation
Understanding the Problem We are given the equation of a line 2 x − 3 y = 6 and asked to find the slope of a line parallel to it. Parallel lines have the same slope, so our goal is to find the slope of the given line.
Plan of Action To find the slope, we will rewrite the equation in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y-intercept.
Isolating y Starting with the given equation 2 x − 3 y = 6 , we want to isolate y on one side of the equation. First, subtract 2 x from both sides: 2 x − 3 y − 2 x = 6 − 2 x
− 3 y = − 2 x + 6
Solving for y Next, divide both sides by − 3 to solve for y :
− 3 − 3 y = − 3 − 2 x + 6
y = − 3 − 2 x + − 3 6
y = 3 2 x − 2
Finding the Slope Now the equation is in slope-intercept form, y = m x + b . We can see that the slope m is 3 2 . Since parallel lines have the same slope, the slope of the line parallel to 2 x − 3 y = 6 is also 3 2 .
Final Answer Therefore, the slope of the line parallel to the line 2 x − 3 y = 6 is 3 2 .
Examples
Understanding slopes is crucial in various real-world applications. For instance, when designing roads or ramps, engineers use the concept of slope to ensure they are not too steep for vehicles or people to navigate safely. Similarly, in architecture, the slope of a roof is carefully calculated to allow for proper water runoff and prevent structural damage. In economics, the slope of a supply or demand curve can tell you how responsive consumers or producers will be to changes in price. These examples show how a simple mathematical concept like slope can have significant practical implications.
