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Feliz [49]
3 years ago
7

The sum of two positive integers, x and y, is not more than 40. The difference of the two integers is at least 20. Chancee choos

es x as the larger number and uses the inequalities y ≤ 40 – x and y ≤ x – 20 to determine the possible solutions. She determines that x must be between 0 and 10 and y must be between 20 and 40. Determine if Chancee found the correct solution. If not, state the correct solution.

Mathematics
1 answer:
kvasek [131]3 years ago
5 0

Answer:

No, Chancee mixed up the variables, the correct solution is that x must be between 20 and 40 and y must be between 0 and 10.

Step-by-step explanation:

Here, x and y are positive integers,

Such that, their sum is not more than 40,  

i.e. x + y ≤ 40 where x, y ≥ 0,

⇒ y ≤ 40 - x                 ...............(1),

Their difference is at least 20,

i.e. x - y ≥ 20

-y ≥ 20 - x

⇒ y ≤ x - 20   ............(2),

Graphing y ≤ 40 - x :

Related equation of inequality (1),

y = 40 - x

if x = 0, y = 40,

if y = 0, 0 = 40 - x ⇒ x = 40,

'≤' represents a solid line,

Join the points (0, 40) and (40, 0) by solid line.

Graphing y ≤ x - 20:

Related equation of inequality (1),

y = x - 20,

if x = 0, y = -20,

if y = 0, 0 = x - 20 ⇒ x = 20,

'≤' represents a solid line,

Join the points (0, -20) and (20, 0) by solid line.

By graphing we obtained a feasible region that is the solution of the system of inequalities (1) and (2) ( shown below ),

In which,

20 ≤ x ≤ 40 and 0 ≤ y ≤ 30

Hence, Chancee did not find the correct solution.

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These problems are called systems of equations. Basically you have two linear equations and you need to find the values for x and y. In other words, all these equation are lines and our answer will be the exact point that the pair of lines intersect. For example, if we get x=1 and y=2 the lines will intersect at point (1,2). Now that you have some background knowledge here comes the tricks and tactics kid.

We know that we can solve one variable equation easily. For example...

x+1=2

x=1 obviously

Cause we have two variables x and y it is not possible to find a solution. For example, in the equation x+y=10, x=1 when y=9 and x=2 when y=8. There is not correct answer.

So what can we do? We have to make a two variable equation into a one variable equation.

There are two ways to do this: substitution and elimination. I will create a sample problem and then solve it using both methods.

x+y=2

2y-y=1

3)  

-3x-5y=-7 -----> -12x-20y=-28

-4x-3y=-2 ------> -12x-9y=-6

 -12x-20y=-28

-(-12x-9y=-6)

---------------------

-11y=-22

y=2

-3x-5(2)=-7

-3x=3

x=-1

4) 8x+4y=12 ---> 24x+12y=36

7x+3y=10 ---> 28x+12y=40

 28x+12y=40

-(24x+12y=36)

---------------------

4x=4

x=1

8(1)+4y=12

4y=4

y=1

5) 4x+3y=-7

-2x-5y=7 ----> -4x-10y=14

    4x+3y=-7

+(-4x-10y=14)

-------------------

-7y=7

y=-1

4x+3(-1)=-7

4x=-4

x=-1

6) 8x-3y=-9 ---> 32x-12y=-36

5x+4y=12 ---> 15x+12y=36

 32x-12y=-36

+(15x+12y=36)

--------------------

47x=0

x=0

8(0)-3y=-9

-3y=-9

y=3

7)-3x+5y=-2

2x-2y=1 ---> x-y=1/2 ----> x=y+1/2

-3(y+1/2)+5y=-2

-3y-1.5+5y=-2

2y=-0.5

y=0.25

2x-2(0.25)=1

2x=1.5

x=0.75

4 0
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