independent practice
solve.
1 two sections of a garden are shaped like identical isosceles triangles. the base of each triangle is 50 feet, and the other two sides are each x feet long. if the combined perimeter of both gardens is 242 feet, find the value of x.
2 mr. carter has a rectangular plot of land that is 525 feet long and y feet wide. he decides to build a fence around the plot. if the perimeter of the plot is 1,504 feet, find the value of y.
3 the diagram shows an artificial lake. when emma jogged twice around the lake, she jogged a distance of 2,700 meters. find the value of x.
4 sara wants to trim a lampshade with braid. the lampshade is shaped like a rectangular prism. the length of the base of the lampshade is 4 inches greater than its width. if the perimeter of the base is 54 inches, find the length of the base.
5 brooke was given a riddle to solve: the sum of two consecutive positive integers is 71. find the two positive integers.
6 the sum of a negative number, 1/4 of the negative number, and 7/16 of the negative number is -13 1/2. what is the negative number?
7 kevin wrote a riddle: a positive number is 5 less than another positive number. six times the lesser number minus 3 times the greater number is 3. find the two positive numbers.
8 at a charity basketball game, 450 tickets were sold to students at a school. the remaining x tickets were sold to the public. the prices of the two types of tickets are shown. when all the tickets were sold, $10,500 was collected. how many tickets were sold to the public?

Question

Accepted Answer

### 1.
# Explanation:
## Step1: Calculate the combined base - length
Each triangle has a base of 50 feet, and there are 2 triangles. So the combined base - length is $2\times50 = 100$ feet.
## Step2: Calculate the combined length of the non - base sides
The combined perimeter of the two triangles is 242 feet. Let the length of each non - base side of each triangle be $x$ feet. There are 4 non - base sides in total. So the combined length of the non - base sides is $4x$ feet. We can set up the equation $4x+100 = 242$.
## Step3: Solve the equation for $x$
Subtract 100 from both sides of the equation: $4x=242 - 100=142$. Then divide both sides by 4: $x=\frac{142}{4}=35.5$ feet.
# Answer:
$x = 35.5$

### 2.
# Explanation:
## Step1: Recall the perimeter formula for a rectangle
The perimeter formula of a rectangle is $P = 2(l + w)$, where $l$ is the length and $w$ is the width. Here, $l = 525$ feet, $w = y$ feet, and $P=1504$ feet. So the equation is $2(525 + y)=1504$.
## Step2: Solve the equation for $y$
First, divide both sides of the equation by 2: $525 + y=\frac{1504}{2}=752$. Then subtract 525 from both sides: $y=752 - 525 = 227$ feet.
# Answer:
$y = 227$

### 3.
# Explanation:
## Step1: Calculate the perimeter of the lake
Emma jogged twice around the lake and the distance is 2700 meters, so the perimeter $P$ of the lake is $\frac{2700}{2}=1350$ meters.
## Step2: Express the perimeter of the lake in terms of $x$
The perimeter of the lake can be expressed as $x + 250+x + 40+3x+x+x=7x + 290$.
## Step3: Solve the equation for $x$
Set up the equation $7x + 290=1350$. Subtract 290 from both sides: $7x=1350 - 290 = 1060$. Then divide both sides by 7: $x=\frac{1060}{7}\approx151.43$ meters.
# Answer:
$x=\frac{1060}{7}\approx151.43$

### 4.
# Explanation:
## Step1: Let the width of the base be $w$ inches
Then the length $l$ of the base is $w + 4$ inches. The perimeter formula for a rectangle is $P = 2(l + w)$. Here, $P = 54$ inches. So $2((w + 4)+w)=54$.
## Step2: Simplify and solve for $w$
First, expand the left - hand side: $2(2w + 4)=54$, then $4w+8 = 54$. Subtract 8 from both sides: $4w=54 - 8 = 46$, and $w=\frac{46}{4}=11.5$ inches.
## Step3: Calculate the length
The length $l=w + 4=11.5+4 = 15.5$ inches.
# Answer:
15.5 inches

### 5.
# Explanation:
## Step1: Let the first positive integer be $n$
Then the next consecutive positive integer is $n + 1$. The sum of the two consecutive positive integers is $n+(n + 1)=71$.
## Step2: Solve the equation for $n$
Combine like terms: $2n+1 = 71$. Subtract 1 from both sides: $2n=71 - 1 = 70$. Then divide both sides by 2: $n = 35$. The other integer is $n + 1=36$.
# Answer:
35 and 36

### 6.
# Explanation:
## Step1: Let the negative number be $x$
The sum of the negative number, $\frac{1}{4}$ of the negative number, and $\frac{7}{16}$ of the negative number is $x+\frac{1}{4}x+\frac{7}{16}x$. We know that $x+\frac{1}{4}x+\frac{7}{16}x=-13\frac{1}{2}=-\frac{27}{2}$.
## Step2: Combine like terms
First, find a common denominator for the left - hand side. The common denominator of 1, 4, and 16 is 16. So $x+\frac{1}{4}x+\frac{7}{16}x=\frac{16x+4x + 7x}{16}=\frac{27x}{16}$.
## Step3: Solve the equation for $x$
Set up the equation $\frac{27x}{16}=-\frac{27}{2}$. Cross - multiply: $27x\times2=-27\times16$. Then $54x=-432$, and $x=-8$.
# Answer:
$-8$

### 7.
# Explanation:
## Step1: Let the lesser positive number be $x$
Then the greater positive number is $x + 5$. The equation based on the problem is $6x-3(x + 5)=3$.
## Step2: Expand and simplify the equation
Expand the left - hand side: $6x-3x-15 = 3$. Combine like terms: $3x-15 = 3$.
## Step3: Solve the equation for $x$
Add 15 to both sides: $3x=3 + 15 = 18$. Then divide both sides by 3: $x = 6$. The greater number is $x + 5=11$.
# Answer:
6 and 11

### 8.
# Explanation:
## Step1: Set up an equation based on the money collected
The money collected from student tickets is $15\times450$ dollars, and the money collected from public tickets is $25x$ dollars. The total money collected is $10500$ dollars. So the equation is $15\times450+25x=10500$.
## Step2: Simplify the left - hand side
First, calculate $15\times450 = 6750$. The equation becomes $6750+25x=10500$.
## Step3: Solve the equation for $x$
Subtract 6750 from both sides: $25x=10500 - 6750 = 3750$. Then divide both sides by 25: $x = 150$.
# Answer:
150

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QUESTION IMAGE

Question

Explanation:

1.

Step1: Calculate the combined base - length

Step2: Calculate the combined length of the non - base sides

Step3: Solve the equation for \(x\)

Step1: Recall the perimeter formula for a rectangle

Step2: Solve the equation for \(y\)

Step1: Calculate the perimeter of the lake

Step2: Express the perimeter of the lake in terms of \(x\)

Step3: Solve the equation for \(x\)

Answer:

2.