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 we need 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.
Converting to Slope-Intercept Form To find the slope, we need to 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 Let's solve the equation 2 x − 3 y = 6 for y . First, subtract 2 x from both sides: − 3 y = − 2 x + 6
Solving for y Now, divide both sides by − 3 :
y = − 3 − 2 x + − 3 6 y = 3 2 x − 2
Finding the Slope From the slope-intercept form y = 3 2 x − 2 , we can see that the slope of the given line is 3 2 . Since parallel lines have the same slope, the slope of the line parallel to the given line is also 3 2 .
Final Answer Therefore, the slope of the line parallel to 2 x − 3 y = 6 is 3 2 .
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 and accessibility. If a builder needs to construct a ramp parallel to an existing one, knowing that parallel lines have the same slope ensures the new ramp will have the same incline, maintaining consistent accessibility. Similarly, in map reading, parallel roads on a map will have the same slope when represented graphically, aiding in navigation and understanding terrain.
