Find the slope of the line whose equation is $6x - 3y = 12$.
Answer (1)
Rewrite the given equation in slope-intercept form.
Isolate the y term: − 3 y = − 6 x + 12 .
Divide both sides by − 3 : y = 2 x − 4 .
Identify the slope as the coefficient of x : 2 .
Explanation
Understanding the Problem We are given the equation of a line 6 x − 3 y = 12 and asked to find its slope. The slope of a line in the form y = m x + b is given by m . To find the slope, we need to rewrite the given equation in slope-intercept form.
Isolating the y term First, we isolate the y term by subtracting 6 x from both sides of the equation: 6 x − 3 y − 6 x = 12 − 6 x − 3 y = − 6 x + 12
Solving for y Next, we divide both sides of the equation by − 3 to solve for y :
− 3 − 3 y = − 3 − 6 x + 12 y = − 3 − 6 x + − 3 12 y = 2 x − 4
Identifying the Slope Now that the equation is in slope-intercept form, y = 2 x − 4 , we can identify the slope as the coefficient of x , which is 2 .
Final Answer Therefore, the slope of the line is 2 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, if you are analyzing the relationship between the number of hours studied and the score on a test, the slope of the line representing this relationship would tell you how much the test score increases for each additional hour of studying. Similarly, in construction, the slope of a ramp determines its steepness, which is essential for accessibility. Knowing how to find the slope from an equation allows you to quickly understand and interpret these relationships.
