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2 years ago

What is the solution to the system of equations below? {2y=x+63x=y+7 (5, 4) (20, 13) (13, 20) (4, 5)

Mathematics

1 answer:

Tcecarenko [31]2 years ago

7 0

Answer:

(4,5)

Step-by-step explanation:

Let's solve your system by substitution.

2y=x+6;3x=y+7

Step: Solve2y=x+6for x:

2y=x+6

2y+−x=x+6+−x(Add -x to both sides)

−x+2y=6

−x+2y+−2y=6+−2y(Add -2y to both sides)

−x=−2y+6

−x

−1

−2y+6

−1

(Divide both sides by -1)

x=2y−6

Step: Substitute2y−6forxin3x=y+7:

3x=y+7

3(2y−6)=y+7

6y−18=y+7(Simplify both sides of the equation)

6y−18+−y=y+7+−y(Add -y to both sides)

5y−18=7

5y−18+18=7+18(Add 18 to both sides)

5y=25

(Divide both sides by 5)

y=5

Step: Substitute 5 for y in x=2y−6:

x=2y−6

x=(2)(5)−6

x=4(Simplify both sides of the equation)

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Answer:

The total number of family members is 21.

Step-by-step explanation:

To solve this problem, we must build the Venn's Diagram of these sets.

I am going to say that:

-The set A represents those that would not go to a park.

-The set B represents those who would not go to a beach.

-The set C represents those who would not go to the family cottage.

The value d represents those who would go to all three places.

We have that:

$A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)$

In which a are those that would only not go to a park, $A \cap B$ are those who would not got to a park or to the beach, $A \cap C$ are those who would not go to a park or to the famili cottage. And $A \cap B \cap C$ are those that would not go to any of these places.

By the same logic, we have:

$B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)$

$C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

This diagram has the following values:

$a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)$

The total number of family members is the sum of all these values:

$T = a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

We start finding the values from the intersection of the three sets

5 would not go to a park or a beach or to the family cottage.

This means that $A \cap B \cap C = 5$

1 would go to all three places. This means that $d = 1$ .

8 would go to neither a park nor the family cottage

This means that:

$A \cap C + (A \cap B \cap C) = 8$

$A \cap C = 3$

8 would go to neither a beach nor the family cottage

$B \cap C + (A \cap B \cap C) = 8$

$B \cap C = 3$

7 would go to neither a park nor a beach

$A \cap B + (A \cap B \cap C) = 7$

$A \cap B = 2$

15 would not go to the family cottage

$C = 15$

$C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

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12 would not go to a beach

$B = 12$

$B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)$

$12 = b + 3 + 2 + 5$

$b = 2$

11 would not go to a park

$A = 11$

$A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)$

$11 = a + 2 + 3 + 5$

$a = 1$

Now, we can find the total number of family members.

$T = a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

$T = 1 + 2 + 4 + 1 + 2 + 3 + 3 + 5$

$T = 21$

The total number of family members is 21.

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