Question

2 meeting constraints
here are some situations that relate two quantities and involve two constraints. for each situation, find the pair of values that meet both constraints, and explain or show your reasoning.
1 a dining hall had a total of 25 tables—some long rectangular tables and some round ones. long tables can seat 8 people. round tables can seat 6 people. on a busy evening, all 190 seats at the tables are occupied.
how many long tables, x, and how many round tables, y, are there?
2 a family bought a total of 16 adult and child tickets to a magic show. adult tickets are $10.50 each and child tickets are $7.50 each. the family paid a total of $141.
how many adult tickets, a, and child tickets, c, did they buy?

Explanation:

1.

Step1: Set up equations

Let $x$ be the number of long - tables and $y$ be the number of round - tables. We have two equations based on the given information:
The total number of tables equation: $x + y=25$, so $x = 25 - y$.
The total number of seats equation: $8x+6y = 190$.

Step2: Substitute $x$ into the second equation

Substitute $x = 25 - y$ into $8x+6y = 190$.
$8(25 - y)+6y=190$.
Expand the left - hand side: $200-8y + 6y=190$.
Combine like terms: $200-(8y - 6y)=190$, so $200 - 2y=190$.

Step3: Solve for $y$

Subtract 200 from both sides: $-2y=190 - 200=-10$.
Divide both sides by $-2$: $y = 5$.

Step4: Solve for $x$

Substitute $y = 5$ into $x = 25 - y$, then $x=25 - 5=20$.

Step1: Set up equations

Let $a$ be the number of adult tickets and $c$ be the number of child tickets.
We have two equations:
The total number of tickets equation: $a + c=16$, so $a=16 - c$.
The total cost equation: $10.5a+7.5c = 141$.

Step2: Substitute $a$ into the second equation

Substitute $a = 16 - c$ into $10.5a+7.5c = 141$.
$10.5(16 - c)+7.5c=141$.
Expand the left - hand side: $168-10.5c+7.5c = 141$.
Combine like terms: $168-(10.5c - 7.5c)=141$, so $168 - 3c=141$.

Step3: Solve for $c$

Subtract 168 from both sides: $-3c=141 - 168=-27$.
Divide both sides by $-3$: $c = 9$.

Step4: Solve for $a$

Substitute $c = 9$ into $a=16 - c$, then $a=16 - 9 = 7$.

Answer:

$x = 20$ long tables, $y = 5$ round tables

2.