1. Solve the system of equations below both algebraically and by graphing. Be sure to show all of your work and state your solution as an ordered pair.
1. The first thing I did was put both equations in the form of
y=mx+b. That way it makes them easy to graph and I will know the slope (m) and
the y intercept (b) of each.
The first equation is already in y=mx+b.
y=1/2x + 5/2 means the slope of this line is m=1/2 and the y
intercept is 5/2.
For the second equation I moved numbers around.
It started off as 3x+2y=1.
First I subtracted 3x from both sides to get 2y=-3x+1.
Next I divided 2 to every term in the equation to get y alone.
You get y=-3/2x+1/2.
Now that this is in y=mx+b form, I know that the slope of this
line is m=-3/2 and the y intercept is 1/2.
Knowing this information it can now be graphed.
I first graphed the line with the slope of 1/2 and y intercept
The line intercepts the y-axis at 5/2 so there should have a dot
at 5/2 or 2.5
Now from there we need a second point to make the line.
If the slope is 1/2 that means we can either go from this point
UP 2 and RIGHT 1 or DOWN 2 and LEFT 1 and can keep doing this from every point.
Next, I did the same thing for the line where slope is -3/2 (or
-1.5) and has a y intercept of 1/2.
I drew the y intercept first on the y line at the 1/2 point
between 0 and 1.
Since it is a negative slope this time we go either DOWN 2 and
RIGHT 3 or UP 2 and LEFT 3 from each point.
The lines both meet up at the point (-1, 2) so
this is the answer!
I set the two equations equal to each other by substituting one
of the variables.
I have y=1/2x+5/2 and y=-3/2x+1/2.
If y equals that, you can replace y with the thing y equals in
the other equation.
You should get 1/2x+5/2=-3/2x+1/2.
Add 3/2x to both sides and you get 4/2x+5/2=1/2.
Then subtract 5/2 from both sides to get 4/2x=-4/2.
Simplify 4/2 by dividing and you get 2x=-2.
Finally, divide by 2 to both sides and you get x=-1.
Now to get the y variable plug x=-1 back into EITHER equation by
replacing wherever it says x with a -1.
I chose to plug in to y=1/2(-1)+5/2.
Multiplying you get y=-1/2+5/2.
Adding you get y=4/2. Dividing that you get y=2.
Then the answer again is in the form of (x,y) so the answer is (-1,
2) It is the same answer both ways.
2. The population of a country is initially 2.5 million people and is increasing by 0.8 million people every year. The country’s annual food supply is initially adequate for 4 million people and is increasing at a constant rate for an additional 0.4 million people per year.
a. Based on these assumptions, in approximately how many years will the country first experience shortages of food?
b. If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? If so, how many years would it take for shortages to occur? If not, explain.
c. If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur? If so, how many years would it take for shortages to occur? If not, explain.
a). I used the
variable ‘x’ in place of the unknown years:
population is 2.5+0.8x where x is 0
supply is 4+0.4x where x= 0
years is when supply will be equal to the demand of the population
the above 3.75 years there will be shortage of food.
off to the next year we have X=4 years this indicates the years that food
shortage will be experienced.
The answer is approximately 4 years.
b. Initial food supply is 4 million
years this is actually the time supply will be equal to the demand of the
above 18.3333 years there will be shortage.
Hence 19 years to come it means that there will be
c. Supply function 2(4+0.4x) =8+0.8x
function is 2.5+0.8x
0=-5.5 and this is a false statement
This means that for the rest of the years food will
The negative (-) 5.5 means that each year will have
an additional food for 5.5 million people although it does not increase.
3. Springfield will be opening a new high school in the fall. The number of underclassmen (9th and 10th graders) must fall between 500 and 600 (inclusive), the number of upperclassmen (11th and 12th graders) must fall between 400 and 500 (inclusive), and the number of students cannot exceed 1000.
a. Let x represent the number of underclassmen and y represent the number of upperclassmen. Write a system of inequalities that models the situation.
b. Graph the solution to the system of inequalities in part a.
3. For this one I graphed the equation on a number line. I made
underclassmen x and upperclassmen y.
The number of underclassmen has to fall between 500 and 600 and
inclusive means that it can actually equal 500 or 600. Represent this
Upperclassmen have to be between 400 and 500. Represent this
It also says the total number of students cannot exceed 1000
which means they have to be less than or equal to 1000.
Represent this with the underclassmen plus the upperclassmen is
less than or equal to 1000 or in equation form x+y</=1000
The three equations should be:
If you know the minimum x can be is 500 and the minimum y can be
is 400 then you know the minimum number of students in the school can be 900. x
and y together could also get as high as 1100 but it tells you that
upperclassmen and underclassmen together cannot exceed 1000. That means the
number of students in the school must be greater than or equal to 900 and less
than or equal to 1000.