To begin, we analyze the equation given: the base equation,  is shifted left one unit and vertically stretched by a factor of 2. The graph of the equation  is:

To solve the inequality, we need to take a test point and plug it in to see if it matches the inequality. The only points that cannot be used are those directly on our parabola, so let's use the origin . If plugging this point in makes the inequality true, then we shade the area containing that point (in this case, outside the parabola); if it makes the inequality untrue, then the opposite side is shaded (in this case, the inside of the parabola). Plugging the numbers in shows:

Simplified as:

Which is not true, so the area inside of the parabola should be shaded, resulting in the following graph:

Let x be the number of pencils and y be the number of pens Sally purchased. We are told that each pencil costs 25 cents. Because, the total amount she spent was given in dollars, we want to convert 25 cents to dollars. Because there are 100 cents in one dollar, 25 cents  = $0.25. Similarly, each pen costs $0.50.

The amount that Sally spent on pencils is equal to the product of the number of pencils and the cost per pencil. We can model the amount of money spent on pencils as 0.25x.

Likewise, the amount Sally spent on pens is equal to the product of the number of pens and the cost per pen, or 0.5y.

Since we are told that Sally spent $5.75 total on pens and pencils combined, we can write the following:

We are also told that the number of pens is one greater than the number of pencils. We can thus write:

We now have two equations and two unknowns. In order to solve this system of equations, we can take the value of y = x + 1 and substitute it into the other equation.

Distribute.

Combine x terms.

Subtract 0.5 from both sides.

Divide both sides by 0.75.

This means Sally bought 7 pencils and 8 pens. The question asks us to find the number of pens and pencils purchased altogether, which would equal

The answer is 15.

First, solve this equation for y and then substitute the answer into the second equation:

Now substitute into the second equation and solve for x:

To solve for x, add the coefficients on the x variables together then divide both sides by three. 

Set  as the number of people and  as the number of dogs.

The head equation is going to be:

The legs equation is going to be:

Manipulating the first equation to solve for  results in the following.

Plugging in

 into the second equation and solving for , you get

We can solve either by substitution or by elimination. We will solve by elimination.

Line up both equations so the variables are in the same order.

Because we have a negative  and a positive , if we were to add these two equations straight up and down, the  values would cancel out.

Solve for .

Plug  back in to one of the original equations to solve for .

The final answer will be . We can check this answer by plugging it back into either of the original equations.

Use the substitution method to solve the system of equations. There are two ways to do this - either substitute the first equation into the second, or the second equation into the first. Since the second equation is already solved for one variable, we will choose the latter.

Now factor the equation:

 or 

For this particular question, you could stop right here, as you can only need to know the number of solutions not the ordered pair of the solutions. Here there are two solutions. The ordered pairs would be:

 and 

Subtracting the second equation from the first, we acquire .

Adding this equation to the third equation, we get .

Therefore, .

Solve for  by merging the equations so that  gets factored out.  To do so, multiply the lower equation by  (so that  at the top is subtracted by )

If the total milkshake profits were $246.05, and vanilla milkshakes cost $1.30 while chocolate and strawberry milkshakes cost $1.45, we can set up the equation 

,

where a represents the number of vanilla milkshakes sold and b represents the number of chocolate and strawberry milkshakes sold. We also know that 179 total milkshakes were sold, so we can set up the equation 

.

If we solve for b, 

and substitute that value into our first equation 

we can solve for a and then b, and we get

This means that 90 vanilla milkshakes were sold and 89 chocolate and strawberry milkshakes were sold. If there were 27 more chocolate milkshakes sold than strawberry, we can set up the equation  where s represents the number of strawberry milkshakes sold, and s+27 represents the number of chocolate milkshakes sold. We can now solve for s. 

Therefore, 31 strawberry milkshakes were sold. 

In order to find the solution, you must set up a system of equations. We know that the product of two numbers, say x and y, is equal to 77.

Therefore, your first equation is

.

Next, we know that one number is 3 less than twice the value of the other number, or that

.

Once we have our two equations, we can use substitution in order to eliminate a variable and solve for the remaining variable.

We know that,

,

so instead of

we can say that (2y - 3) times y equals 77, or

.

To solve for y we need to factor.

becomes

.

Because y must be positive, we know that y must equal 7. Substituting 7 for y into our original equation (xy = 77), we learn that x equals 11.

Therefore  and  and the sum of our two numbers equals 18.