What is the slope of the line represented by the equation $3x + 4y = 8$?
Answer (2)
The slope of the line given by the equation 3 x + 4 y = 8 can be found by rewriting it in slope-intercept form, yielding a slope of − 4 3 .
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Rewrite the given equation in slope-intercept form: y = m x + b .
Isolate the y term: 4 y = − 3 x + 8 .
Divide by the coefficient of y : y = − 4 3 x + 2 .
Identify the slope m as the coefficient of x : − 4 3 .
Explanation
Understanding the Problem We are given the equation of a line: 3 x + 4 y = 8 . Our goal is to find the slope of this line. To do this, 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 the y-term First, we isolate the term with y on one side of the equation: 4 y = − 3 x + 8
Solving for y Next, we divide both sides of the equation by 4 to solve for y : y = 4 − 3 x + 4 8 y = − 4 3 x + 2
Identifying the Slope Now that the equation is in slope-intercept form, y = m x + b , we can identify the slope m . In this case, m = − 4 3 . Therefore, the slope of the line is − 4 3 .
Final Answer The slope of the line represented by the equation 3 x + 4 y = 8 is − 4 3 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For instance, consider a ramp for wheelchair access. The slope of the ramp determines how steep it is. A steeper slope (a larger absolute value) requires more effort to ascend, while a gentler slope (a smaller absolute value) is easier. Similarly, in economics, the slope of a supply or demand curve indicates how responsive the quantity supplied or demanded is to changes in price. A steeper slope means that a small change in price leads to a large change in quantity, and vice versa. In construction, the slope of a roof determines how quickly water or snow will drain off. A steeper slope prevents accumulation and potential damage. These examples illustrate how the concept of slope is fundamental in various fields, enabling us to analyze and design systems effectively.
