Question

name: brea harris
problem variables system solution
sara bought a total of 8 nails at ace hardware. she purchased both 6-penny nails and 10-penny nails. she bought 2 more 6-penny nails than she did 10-penny nails. she had a total of 8 nails. how many 6-penny and 10-penny nails did sara purchase?
x =
y =

lucas has an insect collection that contains only ants and beetles. on tuesday sergio did not have much luck finding insects to add to his collection. the number of ants plus twice the number of beetles was only eight. the difference between the number of ants and beetles was -4. how many ants and how many beetles did he catch?
x =
y =

the difference between a snert and a snart is six. the sum of twice a snert and a snart is eighteen. how large is a snert and how large is a snart?
x =
y =

joe was looking through a numeric puzzle book and found the following puzzle. the sum of two numbers is two. the difference of the two numbers is also two. what are the two numbers?
x =
y =

Explanation:

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Problem 1: Sara's Nails

Step1: Define variables

Let $x$ = number of 6-penny nails, $y$ = number of 10-penny nails

Step2: Set up system of equations

Total nails: $x + y = 8$; 6-penny count: $x = y + 2$

Step3: Substitute $x$ into first equation

$(y + 2) + y = 8$

Step4: Solve for $y$

$2y + 2 = 8 \implies 2y = 6 \implies y = 3$

Step5: Solve for $x$

$x = 3 + 2 = 5$

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Problem 2: Ants and Beetles

Step1: Define variables

Let $x$ = number of ants, $y$ = number of beetles

Step2: Set up system of equations

Total relationship: $x + 2y = 8$; Difference: $x - y = -4$

Step3: Isolate $x$ from second equation

$x = y - 4$

Step4: Substitute $x$ into first equation

$(y - 4) + 2y = 8$

Step5: Solve for $y$

$3y - 4 = 8 \implies 3y = 12 \implies y = 4$

Step6: Solve for $x$

$x = 4 - 4 = 0$

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Problem 3: Snert and Snart

Step1: Define variables

Let $x$ = size of a snert, $y$ = size of a snart

Step2: Set up system of equations

Difference: $x - y = 6$; Sum relationship: $2x + y = 18$

Step3: Isolate $x$ from first equation

$x = y + 6$

Step4: Substitute $x$ into second equation

$2(y + 6) + y = 18$

Step5: Solve for $y$

$2y + 12 + y = 18 \implies 3y = 6 \implies y = 2$

Step6: Solve for $x$

$x = 2 + 6 = 8$

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Problem 4: Numeric Puzzle

Step1: Define variables

Let $x$ = first number, $y$ = second number

Step2: Set up system of equations

Sum: $x + y = 2$; Difference: $x - y = 2$

Step3: Add the two equations

$(x + y) + (x - y) = 2 + 2$

Step4: Solve for $x$

$2x = 4 \implies x = 2$

Step5: Solve for $y$

$2 + y = 2 \implies y = 0$

Answer:

1. Sara's Nails:

Variables: $x$ = 6-penny nails, $y$ = 10-penny nails
System:

$$\begin{cases} x + y = 8 \\ x = y + 2 \end{cases}$$

Solution: 5 six-penny nails, 3 ten-penny nails

1. Ants and Beetles:

Variables: $x$ = ants, $y$ = beetles
System:

$$\begin{cases} x + 2y = 8 \\ x - y = -4 \end{cases}$$

Solution: 0 ants, 4 beetles

1. Snert and Snart:

Variables: $x$ = snert size, $y$ = snart size
System:

$$\begin{cases} x - y = 6 \\ 2x + y = 18 \end{cases}$$

Solution: Snert = 8, Snart = 2

1. Numeric Puzzle:

Variables: $x$ = first number, $y$ = second number
System:

$$\begin{cases} x + y = 2 \\ x - y = 2 \end{cases}$$

Solution: 2 and 0