Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. The river was 60 feet before the drought began. Write an equation in standard form
Let x represent the number of days of no rain.
The equation is y = mx + b, where m is the slope and b is the y-intercept.
We can find the slope, m, by using the slope formula: m = (y2 - y1)/(x2 - x1), where (x1, y1) is (3, 45) and (x2, y2) is (10, 10).
m = (10 - 45)/(10 - 3) = -35/7 = -5
Substituting the slope and the point (3, 45) into the equation, we get: y = -5x + b
45 = -5(3) + b
45 = -15 + b
b = 45 + 15 = 60
Therefore, the equation in standard form is 5x + y = 60.
Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. Given the two points we know: (3,45) and (10,10), what would the point slope equation look like?
The point-slope form of equation is given by: y - y1 = m(x - x1)
Using the points (3, 45) and (10, 10), we can find the slope (m) using the formula:
m = (y2 - y1)/(x2 - x1),
where (x1, y1) is (3, 45) and (x2, y2) is (10, 10).
m = (10 - 45)/(10 - 3) = -35/7 = -5
Choose the point (3, 45) to substitute into the point-slope equation:
y - 45 = -5(x - 3)
Simplifying the equation:
y - 45 = -5x + 15
Finally, we can rewrite the equation in slope-intercept form by isolating y:
y = -5x + 60
Therefore, the point-slope equation of the line with the given points is y = -5x + 60.
To write an equation in standard form, we need to find the slope-intercept form of the equation first and then convert it into standard form.
Let's denote the number of days without rain as 'x' and the water level as 'y'. We are given two points on the graph: (3, 45) and (10, 10).
Using the slope-intercept form of an equation, y = mx + b, where 'm' is the slope and 'b' is the y-intercept, we can find the equation in slope-intercept form.
First, let's find the slope (m):
m = (change in y) / (change in x)
m = (10 - 45) / (10 - 3)
m = -35 / 7
m = -5
Now, substitute the slope and one of the points (3, 45) into the slope-intercept form:
45 = -5(3) + b
Simplifying this equation, we get:
45 = -15 + b
b = 45 + 15
b = 60
Therefore, the equation in slope-intercept form is:
y = -5x + 60
To convert this equation into standard form, Ax + By = C, we need to eliminate any fractions and make sure that 'A' is a positive integer.
Multiplying the entire equation by -1, we get:
-y = 5x - 60
Rearranging the terms, we have:
5x + y = 60
Now, we just need to make sure 'A' is a positive integer. We can multiply the entire equation by 5:
25x + 5y = 300
Therefore, the equation in standard form is:
25x + 5y = 300
To write an equation in standard form, we need to use the slope-intercept form of a linear equation, which is y = mx + b, where y is the dependent variable (in this case, the water level), x is the independent variable (in this case, the number of days without rain), m is the slope, and b is the y-intercept.
First, let's determine the change in the water level over time. From day 3 to day 10, the water level decreased by 35 feet (45 - 10 = 35). So, in 7 days, the water level decreased by 35 feet. This gives us a slope of -5 feet per day (-35 รท 7 = -5).
Next, we can use the slope-intercept form to determine the y-intercept. We know that the water level was 60 feet before the drought began (at x = 0). So, b = 60.
Now we can write the equation in slope-intercept form:
y = -5x + 60
To convert this equation to standard form, we need to rearrange it to have the x and y terms on the same side and the coefficient of x to be positive. So, let's rewrite the equation:
5x + y = 60
Therefore, the equation in standard form is 5x + y = 60.