Respuesta :
Problem 10
The error occurs in line 2. Notice how in the previous line, the variable d doesn't have an x attached to it. So we can't factor out x from it if it doesn't exist there.
Instead, the expression ax+bx+d would become x(a+b)+d and not x(a+b+d). You can verify this by distributing the x back through.
Put another way: if we distributed the x in x(a+b+d), then we'd get ax+bx+dx; however there isn't supposed to be an x attached to the d.
Here's how we should solve for x.
[tex]k = ax+bx+d\\\\k-d = ax+bx\\\\k-d = x(a+b)\\\\x = \frac{k-d}{a+b}[/tex]
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Problem 11
Answers:
- a) infinitely many solutions
- b) no solutions
- c) no solutions
- d) infinitely many solutions
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Explanation:
Any time we have the same number on both sides, this means we have infinitely many solutions. If both sides are different, then we have no solutions. For either case, this only applies once the variables are completely gone.
Consider something like x+x = 2x. Both sides turn into 2x and we can subtract 2x from both sides to get 0 = 0. No matter what we plug in for x, it will be a true statement. Therefore, this equation has infinitely many solutions.
For an example that has no solutions, try something like 2x+1 = 2x. Subtracting 2x from both sides yields 1 = 0 which is always false. So 2x+1 = 2x is always false regardless of what you pick for x.
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Problem 12
Answers:
- 3.5 hours driving to the delivery site
- 5.5 hours driving back
Both of those values are exact.
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Explanation:
Let's say point A is the starting point and B is the delivery point. The truck goes from A to B, then back to A again. Let's call the one-way distance d. So we can think of it like saying "the length of segment AB is d miles" even though technically the two cities aren't connected with a straight line road.
When going from A to B, the speed is 55 mph. The time it takes to drive this is
[tex]\text{distance} = \text{rate}*\text{time}\\\\d = r*t\\\\d = 55t\\\\t = \frac{d}{55}\\\\[/tex]
Through similar logic, the time to drive back at 35 mph is [tex]t = \frac{d}{35}\\\\[/tex] because the truck driver is driving the same distance (assume she takes the same roads).
The total time spent driving is [tex]\frac{d}{55}+\frac{d}{35}[/tex]. This total drive time is 9 hours, which means we have this equation we need to solve
[tex]\frac{d}{55}+\frac{d}{35} = 9[/tex]
Let's multiply both sides by the LCD 385 to clear out the denominators
That will help us isolate the variable d.
[tex]\frac{d}{55}+\frac{d}{35} = 9\\\\385*\left(\frac{d}{55}+\frac{d}{35}\right) = 385*9\\\\385*\left(\frac{d}{55}\right)+385*\left(\frac{d}{35}\right) = 3465\\\\7d+11d = 3465\\\\18d = 3465\\\\d = 3465/18\\\\d = 192.5\\\\[/tex]
The distance from A to B is exactly 192.5 miles.
We can then say:
- d/55 = (192.5)/55 = 3.5 which is exact
- d/35 = (192.5)/35 = 5.5 which is also exact
These two results are time values in hours.
So it takes exactly 3.5 hours to go from A to B. Then it takes exactly 5.5 hours to go from B to A. The second leg of the trip is longer in duration as expected due to the slower average speed.
As a check,
3.5+5.5 = 9
which confirms our answers.
