Question

1. the sum of two numbers is 15. twice the first number added to three times the second number is 36. write a system of linear equations to represent this information and find the values of the two numbers.
2. jane is organizing a fundraiser. the total number of adults and children attending is 80. the entrance fee for adults is $10, and for children, it is $5. the total amount collected is $600. write a system of linear equations to represent this information and find the number of adults and children.
3. alice has $50 and decides to save $5 each week. bob has no savings initially but saves $8 each week. after how many weeks will alice and bob have the same amount of money saved?
4. the present ages of two friends are such that the sum of their ages is 50. six years ago, one friends age was twice the age of the other. determine their present ages.

Explanation:

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Problem 5

Step1: Define variables

Let the first number be $x$, the second be $y$.

Step2: Set up equations

Sum: $x + y = 15$
Combined total: $2x + 3y = 36$

Step3: Solve first equation for $x$

$x = 15 - y$

Step4: Substitute into second equation

$2(15 - y) + 3y = 36$
$30 - 2y + 3y = 36$
$30 + y = 36$

Step5: Solve for $y$

$y = 36 - 30 = 6$

Step6: Find $x$

$x = 15 - 6 = 9$

Step1: Define variables

Let adults = $a$, children = $c$.

Step2: Set up equations

Total attendees: $a + c = 80$
Total funds: $10a + 5c = 600$

Step3: Simplify second equation

Divide by 5: $2a + c = 120$

Step4: Subtract first equation

$(2a + c) - (a + c) = 120 - 80$
$a = 40$

Step5: Find $c$

$c = 80 - 40 = 40$

Step1: Define variables

Let weeks = $w$, savings = $S$.

Step2: Set up savings equations

Alice: $S = 50 + 5w$
Bob: $S = 8w$

Step3: Set equations equal

$50 + 5w = 8w$

Step4: Solve for $w$

$50 = 8w - 5w$
$50 = 3w$
$w = \frac{50}{3} \approx 16.67$

Answer:

The system of equations is

$$\begin{cases}x + y = 15 \\ 2x + 3y = 36\end{cases}$$

. The two numbers are 9 and 6.

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Problem 6