State the intercepts and slope of the line. Write intercepts as ordered pairs.
[tex]7 x+46 y=7[/tex]
x-intercept: $\square$
y -intercept: $\square$
Slope: $\square$
Answer (1)
Set y = 0 to find the x-intercept: 7 x = 7 , so x = 1 . The x-intercept is ( 1 , 0 ) .
Set x = 0 to find the y-intercept: 46 y = 7 , so y = 46 7 . The y-intercept is ( 0 , 46 7 ) .
Rewrite the equation in slope-intercept form: y = 46 − 7 x + 46 7 . The slope is 46 − 7 .
State the intercepts and slope: x-intercept: ( 1 , 0 ) , y-intercept: ( 0 , 46 7 ) , slope: − 46 7 .
Explanation
Problem Analysis We are given the equation of a line: 7 x + 46 y = 7 . Our goal is to find the x-intercept, y-intercept, and the slope of this line. Let's tackle each part step by step.
Finding the x-intercept To find the x-intercept, we set y = 0 in the equation and solve for x . This is because the x-intercept is the point where the line crosses the x-axis, and on the x-axis, the y-coordinate is always 0. So, we have:
7 x + 46 ( 0 ) = 7 7 x = 7 x = 7 7 = 1
Therefore, the x-intercept is the point ( 1 , 0 ) .
Finding the y-intercept To find the y-intercept, we set x = 0 in the equation and solve for y . This is because the y-intercept is the point where the line crosses the y-axis, and on the y-axis, the x-coordinate is always 0. So, we have:
7 ( 0 ) + 46 y = 7 46 y = 7 y = 46 7
Therefore, the y-intercept is the point ( 0 , 46 7 ) .
Finding the Slope To find the slope, we can rewrite the equation in the slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting with the given equation:
7 x + 46 y = 7 46 y = − 7 x + 7 y = 46 − 7 x + 46 7
From this, we can see that the slope m = 46 − 7 .
Final Answer In summary:
The x-intercept is ( 1 , 0 ) .
The y-intercept is ( 0 , 46 7 ) .
The slope is 46 − 7 .
Examples
Understanding intercepts and slopes is crucial in many real-world applications. For example, imagine you're tracking the water level in a tank. The y-intercept might represent the initial water level, and the slope could indicate the rate at which the water level is changing over time (filling or draining). By knowing these parameters, you can predict future water levels and manage the tank effectively. Similarly, in economics, the slope of a cost function can represent the marginal cost of production, while the y-intercept represents the fixed costs. Analyzing these values helps businesses make informed decisions about pricing and production levels.
