$y=10-\frac{2}{3} x$
Answer (1)
The equation is in slope-intercept form, y = m x + b .
Identify the slope: m = − 3 2 .
Identify the y-intercept: b = 10 .
Calculate the x-intercept by setting y = 0 : x = 15 .
The line has a slope of − 3 2 , a y-intercept of 10 , and an x-intercept of 15 .
Explanation
Problem Analysis The problem provides the equation of a line: y = 10 − 3 2 x . However, it does not specify what we need to do with this equation. It could be to graph it, find its slope and y-intercept, find points on the line, or something else. Without a clear objective, I will assume the goal is to analyze the equation and determine its key properties.
Identifying Slope and Y-Intercept The given equation is in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Comparing y = 10 − 3 2 x to y = m x + b , we can identify the slope and y-intercept.
Determining Slope and Y-Intercept The slope m is the coefficient of x , which is − 3 2 . The y-intercept b is the constant term, which is 10 . Therefore, the line has a slope of − 3 2 and a y-intercept of 10 .
Finding the X-Intercept To find the x-intercept, we set y = 0 and solve for x :
0 = 10 − 3 2 x
3 2 x = 10
x = 10 × 2 3
x = 15
So, the x-intercept is 15 .
Summary of Key Properties In summary, the line has a slope of − 3 2 , a y-intercept of 10 , and an x-intercept of 15 .
Examples
Understanding the slope and intercepts of a line is crucial in many real-world applications. For example, if this equation represents the budget line for a consumer, the slope indicates the rate at which one good can be exchanged for another, and the intercepts show the maximum amount of each good that can be purchased. Similarly, in physics, this could represent the relationship between distance and time for an object moving at a constant velocity, where the slope is the velocity and the y-intercept is the initial position.
