The break-even point is where the costs equal revenue.

Let \(\displaystyle w\) = # of widgets sold.

Costs:  \(\displaystyle C(w) = 45w + 750\)

Revenue:  \(\displaystyle R(w) = 75w\)

So the equation to solve becomes \(\displaystyle 45w + 750 = 75w\)

So the break-even point occurs when they sell 25 widgets.

Profits = Revenues - Costs

Revenue:  \(\displaystyle R(w) = 75w\)

Costs:  \(\displaystyle C(w) = 45w + 750\)

Profit: \(\displaystyle P(w) = 75w - (45w + 750) = 30w -750\)

So the equation to solve becomes \(\displaystyle 3,000 = 30w - 750\)

So a $3,000 profit occurs when they sell 125 widgets

Let \(\displaystyle x\) = # of frames sold

\(\displaystyle Profits = Revenues-Costs\)

Revenues:  \(\displaystyle R(x) = 35x\)

Costs:  \(\displaystyle C(x) = 10x + 350\)

Profits = \(\displaystyle P(x) = R(x) - C(x) = 35x - (10x + 350) = 25x - 350\)

So the equation to solve becomes \(\displaystyle 500 = 25x - 350\)

So 34 picture frames must be sold to make a $500 profit.

Let pure water be 0% and pure solution be 100%.

So the general equation to solve is:

\(\displaystyle V_{1}P_{1} + V_{2}P_{2} = V_{f}P_{f}\) where \(\displaystyle V\) is the volume and the \(\displaystyle P\) is percent solution.

So the equation to solve becomes \(\displaystyle x(0) + 2(0.90) = (x + 2)(0.30)\)

Solving shows that we need to add 4 gallons of pure water to 2 gallons of 90% pure cleaning solution to get a 30% pure solution.

Let \(\displaystyle x\) = number of dimes, \(\displaystyle x + 1\) = number of nickels, and 

\(\displaystyle (x +1) - 2 = x - 1\) = number of quarters.

The general equation to use is:

\(\displaystyle V_{1}N_{1} + V_{2}N_{2} + V_{3}N_{3} = V_{f}\) where \(\displaystyle V\) is the money value and \(\displaystyle N\) is the number of coins

So the equation to solve becomes

\(\displaystyle 0.10x + 0.05(x + 1) + 0.25(x - 1) = 1.40\)

Thus, solving the equation shows that she had five nickels, four dimes, and three quarters giving a total of 12 coins.

Pure water is 0% and pure solution is 100%

\(\displaystyle V_{1}P_{1} + V_{2}P_{2} = V_{f}P_{f}\) where \(\displaystyle V\) is the volume and \(\displaystyle P\) is the percent.

So the equation to solve becomes \(\displaystyle x(0)+1(0.80)=(1+x)(0.25)\)

So we need to add \(\displaystyle 2.2\ L\) pure water to \(\displaystyle 1\ L\) of 80% cleaning solution to yield 25% cleaning solution.

We are looking for the value of t  that gives $675 as the result when plugged in V (t ). While there are many ways to do this, one of the fastest is to plug in the answer choices as values of t .

When we plug t  = 1 into V (t ), we get V (1) = 1200(0.75)¹ = 1000(0.75) = $900, which is incorrect.

When we plug t  = 2 into V (t ), we get V (2) = 1200(0.75)² = $675, so this is our solution.

The value of the tractor will be $675 after 2 years.

Finally, we can see that if t  = 3, 4, or 5, the resulting values of the V (t ) are all incorrect.

First combine like terms. In this case, 4x and 9x can be added together:

13x + 13 = 0

Subtract 13 from both sides:

13x = -13

Divide both sides by 13 to isolate x:

x = -13/13

x = -1

The total weight of the first three dogs is 225 pounds. This amount, plus the weight of the fourth dog, divided by total number of dogs, is the new average weight:

\(\displaystyle \frac{d + 225}{4} = 70\)

\(\displaystyle d + 225 = 280\)

\(\displaystyle d = 55 lbs\)

Let \(\displaystyle x=\) the number of bird houses sold each month.

\(\displaystyle Revenue = 40x\)

\(\displaystyle Costs=15x+750\)

The break-even point is where the revenue is the same as the costs:

\(\displaystyle Revenue=Costs\) 

\(\displaystyle 40x=15x+750\) 

Solve for \(\displaystyle x\):

\(\displaystyle x=30\)

Therefore, Pets Plus must sell 30 bird houses to break-even.