please write and equation, show your work, and find the answer:
1.a certain two-digit number has a value that is seven more than six times the sum of its digits. the tens digit is 3 more than the units digit. find the number
2.In Colorado Creek, Darrell can row 24km downstream in 6 hours or he can row 18km upstream in the same amount of time. find the rate he rows in still water and the rate of the current.
3. if the temperature on a Celsius scale is divided by five it is equal to the reading on a Fahrenheit scale. fifteen times the Fahrenheit reading is equal to three times the Celsius reading. please find the temperature on each scale.
2.) distance = time × speed
Let rowing speed in still water be x, current in river be y
Downstream, speed is his natural rowing speed plus the current:
24 = 6(x + y)
Upstream, speed is natural rowing speed minus current:
18 = 6(x - y)
or
4 = x + y
3 = x - y
Add
7 = 2 x
x = 7/2 km/h (and current is 1/2 km/h)
He can row at 3.5 km/h
1. Let's say that the tens digit is represented by a variable "x" and the units digit is represented by a variable "y".
According to the given information, the value of the two-digit number is seven more than six times the sum of its digits:
10x + y = 6(x + y) + 7
The tens digit is also three more than the units digit:
x = y + 3
To find the number, we need to solve this system of equations. Let's substitute the value of x from the second equation into the first equation:
10(y + 3) + y = 6(y + 3 + y) + 7
10y + 30 + y = 6(2y + 3) + 7
11y + 30 = 12y + 18 + 7
11y - 12y = 25 - 30
-y = -5
y = 5
Now substitute this value of y back into the equation x = y + 3:
x = 5 + 3
x = 8
So, the number is 85.
2. Let's represent the rate Darrell rows in still water as "r" and the rate of the current as "c".
The distance formula for rowing downstream is:
Distance = Rate * Time
24 = (r + c) * 6
The distance formula for rowing upstream is:
Distance = Rate * Time
18 = (r - c) * 6
Now we have a system of equations. Let's solve for r and c.
From the first equation:
24 = (r + c) * 6
4 = r + c
From the second equation:
18 = (r - c) * 6
3 = r - c
Now we have a system of linear equations:
r + c = 4
r - c = 3
Solve this system of equations using any method (elimination, substitution, etc.) to find the values of r and c.
Adding the two equations eliminates "c":
2r = 7
r = 7/2
Substituting this value back into one of the equations (e.g., r + c = 4):
(7/2) + c = 4
c = 4 - (7/2)
c = 8/2 - 7/2
c = 1/2
Therefore, the rate Darrell rows in still water is 7/2 km/h and the rate of the current is 1/2 km/h.
3. Let's say that the temperature on the Celsius scale is represented by "C" and the temperature on the Fahrenheit scale is represented by "F".
According to the given information, if the temperature on the Celsius scale is divided by five, it is equal to the reading on the Fahrenheit scale:
C/5 = F
Also, fifteen times the Fahrenheit reading is equal to three times the Celsius reading:
15F = 3C
We have a system of equations here. To solve it, we can use substitution or elimination.
Let's solve it by substitution. Rearrange the first equation to isolate C:
C = 5F
Substitute this value of C into the second equation:
15F = 3(5F)
15F = 15F
This equation tells us that F can be any value. So, there are infinitely many solutions to this problem. If we choose a value for F, we can find the corresponding value for C using the first equation.
For example, let's choose F = 10. Plugging this into the equation C = 5F:
C = 5 * 10
C = 50
Therefore, when the temperature on the Fahrenheit scale is 10°F, the temperature on the Celsius scale is 50°C.