# Solving Equations with Parentheses and Finding Areas of Rectangles: What is Heavy Forward But Not Backward Worksheet Answers PDF

## What is heavy forward but not backward worksheet answers pdf

If you are looking for a fun and challenging way to practice your algebra skills, you might want to try this worksheet that asks you to solve equations with parentheses and find the area of a rectangle. The worksheet also contains a riddle that you can solve by using your answers to these problems. In this article, we will explain how to solve equations with parentheses, how to find the area of a rectangle, and how to answer the riddle "what is heavy forward but not backward?"

## what is heavy forward but not backward worksheet answers pdf

## How to solve equations with parentheses

Equations with parentheses are also known as equations with grouping symbols. They require you to apply some special rules when solving them. Here are the steps you need to follow:

Step 1: Simplify the expressions inside the parentheses by applying the order of operations. This means that you have to perform any calculations involving exponents, multiplication, division, addition, and subtraction in that order.

Step 2: Use the distributive property to remove the parentheses and combine like terms. The distributive property states that a(b + c) = ab + ac, which means that you can multiply a term outside the parentheses by each term inside the parentheses and add them together.

Step 3: Isolate the variable by adding or subtracting terms from both sides of the equation. A variable is a letter that represents an unknown number. To isolate it, you have to move all the terms that contain the variable to one side of the equation and all the terms that do not contain the variable to the other side.

Step 4: Divide both sides of the equation by the coefficient of the variable to get the final answer. The coefficient is the number that is multiplied by the variable. For example, in 5x, 5 is the coefficient of x. To get rid of the coefficient, you have to divide both sides of the equation by it.

## How to find the area of a rectangle

A rectangle is a four-sided shape that has two pairs of parallel and equal sides. The area of a rectangle is the amount of space inside it. To find the area of a rectangle, you need to know its length and width. Here are the steps you need to follow:

Step 1: Identify the length and width of the rectangle. The length is the longer side and the width is the shorter side. Sometimes, you might be given one of these measurements and asked to find the other.

Step 2: Multiply the length and width to get the area. The formula for finding the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width.

Step 3: Write an equation that relates the area, length, and width of the rectangle. You can use the formula from step 2 and substitute the values that you know or are given.

Step 4: Solve for the unknown variable by using inverse operations. Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division. For example, if you have to find x in x + 5 = 10, you can use subtraction as an inverse operation and subtract 5 from both sides to get x = 5.

## Examples of solving equations with parentheses and finding the area of a rectangle

Let's look at some examples of how to apply these steps to solve equations with parentheses and find the area of a rectangle.

### Example 1: Solve x + (3x + 4) = 20 and find x.

This is an equation with parentheses that we need to solve for x. Here are the steps:

Simplify the expression inside the parentheses by applying the order of operations. There are no exponents, multiplication, or division in this expression, so we only have to perform addition. We get x + (3x + 4) = 20.

Use the distributive property to remove the parentheses and combine like terms. We multiply x by each term inside the parentheses and add them together. We also combine any terms that have the same variable or are constants. We get 4x + 4 = 20.

Isolate x by adding or subtracting terms from both sides of the equation. We subtract 4 from both sides to move it away from x. We get 4x = 16.

Divide both sides by the coefficient of x to get the final answer. We divide both sides by 4 to get rid of it. We get x = 4.

The answer is x = 4.

### Example 2: Solve 5(k - 3) - 8k = -34 and find k.

This is another equation with parentheses that we need to solve for k. Here are the steps:

Simplify the expression inside the parentheses by applying the order of operations. There are no exponents, multiplication, or division in this expression, so we only have to perform subtraction. We get 5(k - 3) - 8k = -34.

Use the distributive property to remove the parentheses and combine like terms. We multiply 5 by each term inside the parentheses and subtract them from -8k. We also combine any terms that have the same variable or are constants. We get -3k - 15 = -34.

Isolate k by adding or subtracting terms from both sides of the equation. We add 15 to both sides to move it away from k. We get -3k = -19.

Divide both sides by the coefficient of k to get the final answer. We divide both sides by -3 to get rid of it. We get k = 6.33.

The answer is k = 6.33.

### Example 3: Solve for x if the area of the rectangle is 55.5 square units and the length is x + 5.

This is a problem that involves finding the area of a rectangle and solving for x. Here are the steps:

Identify the length and width of the rectangle. The length is given as x + 5 and the width is not given. We can call it w.

Multiply the length and width to get the area. The formula for finding the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width. We substitute the values that we know or are given and get 55.5 = (x + 5)w.

Write an equation that relates the area, length, and width of the rectangle. We already have an equation from step 2, so we can use it as it is.

Solve for x by using inverse operations. We have to isolate x from the other terms in the equation. We can do this by dividing both sides by w to get rid of it. We get 55.5/w = x + 5. Then, we subtract 5 from both sides to move it away from x. We get 55.5/w - 5 = x.

The answer is x = 55.5/w - 5.

### Example 4: Solve for x if the area of the rectangle is 70 square units and the width is x - 2.

This is another problem that involves finding the area of a rectangle and solving for x. Here are the steps:

Identify the length and width of the rectangle. The width is given as x - 2 and the length is not given. We can call it l.

Multiply the length and width to get the area. The formula for finding the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width. We substitute the values that we know or are given and get 70 = l(x - 2).

Write an equation that relates the area, length, and width of the rectangle. We already have an equation from step 2, so we can use it as it is.

Solve for x by using inverse operations. We have to isolate x from the other terms in the equation. We can do this by dividing both sides by l to get rid of it. We get 70/l = x - 2. Then, we add 2 to both sides to move it away from x. We get 70/l + 2 = x.

The answer is x = 70/l + 2.

## What is heavy forward but not backward?

Now that we have solved some equations with parentheses and found some areas of rectangles, we can try to answer the riddle that is on the worksheet: "what is heavy forward but not backward?" To do this, we have to look at the shaded areas on the worksheet that contain our answers to these problems. Each shaded area corresponds to a letter in the alphabet. For example, if our answer is in the shaded area labeled A, then we write down A as part of our answer to the riddle.

If we follow this method, we will get a word that answers the riddle. The word is TON. A ton is a unit of weight that is heavy forward but not backward. If we reverse the word TON, we get NOT, which means nothing or zero, which is not heavy at all.

## Conclusion

In this article, we have learned how to solve equations with parentheses and how to find the area of a rectangle. These are important skills for learning algebra and geometry. We have also solved a riddle that uses our answers to these problems in a creative way. Solving riddles can help us improve our logical thinking and problem-solving skills.

Here are some tips for practicing algebra skills:

Review the order of operations and the distributive property regularly.

Practice solving different types of equations with parentheses, such as ones that involve fractions, decimals, or negative numbers.

Check your answers by plugging them back into the original equation and seeing if they make sense.

Practice finding the area of different shapes, such as triangles, circles, or trapezoids.

Try to create your own riddles using algebra and geometry concepts.

## FAQs

### Q1: What are some other types of equations that involve parentheses?

A1: Some other types of equations that involve parentheses are ones that have more than one pair of parentheses, such as 2(x + 3) + 5(y - 2) = 10, or ones that have nested parentheses, such as 3(2x - (4 - x)) = 12.

### Q2: How can I check if my answer is correct when solving equations with parentheses?

A2: One way to check if your answer is correct is to plug it back into the original equation and see if it makes the equation true. For example, if you solved x + (3x + 4) = 20 and got x = 4, you can check by substituting 4 for x and simplifying. You should get 20 = 20, which is true.

### Q3: What are some real-life applications of finding the area of a rectangle?

A3: Some real-life applications of finding the area of a rectangle are measuring the size of a room, calculating the amount of paint or carpet needed to cover a floor, or estimating the amount of land or crops in a field.

### Q4: What are some other shapes that have formulas for finding their areas?

A4: Some other shapes that have formulas for finding their areas are triangles, circles, trapezoids, parallelograms, and rhombuses. For example, the formula for finding the area of a triangle is A = (1/2)bh, where A is the area, b is the base, and h is the height.

### Q5: What are some other riddles that can be solved using algebra?

A5: Some other riddles that can be solved using algebra are:

What has a face and two hands but no arms or legs? (A clock)

What can you break, even if you never pick it up or touch it? (A promise)

What word begins and ends with an E but only has one letter? (An envelope)

What invention lets you look right through a wall? (A window)

What has many keys but usually only one or two locks? (A keyboard)

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