Expert: Scott A Wilson - 7/11/2010

Question
Mr. Scott A Wilson,
I am having trouble figuring out how to solve these systems. The examples in my book are not very Clear. I was hoping you can help me.
It states: Solve the following systems of equations. If the systems cannot be solved, state that it cannot be solved and why.

1)5x-2y=-24
-4y=-48-10x

2) 4x+5y=-5
  3x+y=21

3) 3x+y=7
  3x+y=10


then i have these where you have to graph the systems

1) graph the system x=7-2y
                   y=3x

and determine the solution set.

im not quite sure how to do that.... how do i graph it to begin with?

2)graph 3x +y<5
same with this one



with this next one, it states: the domain of the function f:
--> x^3-5 is D = {0,1,2,3,4) find the range of f

thank you for your time and effort.i really do appreciate it

Answer
1) The equations are 5x-2y=-24 and -4y=-48-10x.
Now since the 1st equation has a -2y and we are given -4y, divide equation 2 by 2.
This gives -2y=-24-5x.

Put this into the first equation and the result is 5x-5x-24=-24, and this is true for all x.
Thus, the only requirement for the solution is that it is on the line found by dividing the
2nd equation by -4.  That is, y = 12 + 5x/2.

2) Since the 2nd eqation converts to y = 21 - 3x, put that in the 1st equation.
Since there is a 5y in the 1st equation, note that 5y = 105 - 15x.
Put the constants on one side and the x terms on the other, giving ax = b for some constants a,b.
The result for x is then x = b/a.
Put that back in one equation to find y, and then put x and y in the other equation to check.

3) The left side is the same in both equation and the right side is different.
This means there is no solution.

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1) Convert x = 7 - 2y to y = mx + b.
Adding 2y to both sides and subtracting x from both sids almost gives this.
Dividing the resulting equation by 2 does gives this.
The slope is m and the y intercept is b.
Put in x=0 and x=1 to get y values.
Plot both points and two points can be used to determine a line.

For y=3x, the slope is 3 and the y intercept is 0.
This means it crosses at the origin and has slope 3.
Two of the points would then be (0,0) and (1,3), and two points make a line.

To find the solution, put in the 2nd equation for y into the 1st.  This gives x = 7 - 6x.
Add 6x to both sides and then divide by 7 to get the x answer at the intercept.
Put this back into y=3x to get the y value.

Once you have x and y, put them both into x = 7 - 2y to check and make sure they are right.

Since that is the same as y < 5 - 3x, the region is defined to be y = -3x + 5 and all of the are below that line.  The y=intercept is at 5 and the slope of the line is -3, so one point is (0,5).
Another point found by x=1 and is (1,2).  The line can be drawn with a ruler.

Again, the solution is that line and the area below that line.

The range of x³-5 is found by putting {0,1,2,3,4} in for x.
Note that for x = 0,1,2,3, and 4, x³=0,1,6,27, and 64.
All that is left to do is to subtract 5 from each of the numbers.