Find the slope and the $y$-intercept of the line.
$2 x-3 y=9$
(Please provide ONLY the SLOPE, as a decimal rounded to three decimal places)
Answer (1)
Rewrite the given equation 2 x − 3 y = 9 in slope-intercept form y = m x + b .
Isolate y to get y = 3 2 x − 3 .
Identify the slope as m = 3 2 .
Convert the slope to a decimal rounded to three decimal places: 0.667 .
Explanation
Problem Analysis We are given the equation of a line: 2 x − 3 y = 9 . Our goal is to find the slope of this line and express it as a decimal rounded to three decimal places.
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 represents the slope and b represents the y -intercept.
Isolating y Let's isolate y in the given equation:
2 x − 3 y = 9
Subtract 2 x from both sides:
− 3 y = − 2 x + 9
Divide both sides by − 3 :
y = − 3 − 2 x + − 3 9
Simplify:
y = 3 2 x − 3
Identifying the Slope Now we can identify the slope m from the slope-intercept form y = m x + b . In this case, m = 3 2 .
Converting to Decimal To express the slope as a decimal rounded to three decimal places, we divide 2 by 3:
3 2 ≈ 0.667
Final Answer Therefore, the slope of the line is approximately 0.667 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, if you're analyzing the relationship between the number of hours studied and the score on a test, the slope of the line representing this relationship tells you how much the test score increases for each additional hour of studying. Similarly, in economics, the slope of a supply or demand curve indicates how the quantity supplied or demanded changes with a change in price. Knowing the slope helps in making informed decisions and predictions in various fields.
