Question

a rectangular is 42 feet longer than it is wide. its perimeter is 276 fee. (a) let w. be the width of the garden and L. be the length. write a system of linear equations whose solution gives the length and width of the garden.
(b) use the method of substitution to solve the system.

Answer 1

Are you interested in the way to solve the problem or just the answer?

Answer 2

both please if possible so i don't have to keep bugging people with the same problem over and over lol

Answer 3

Great Attitude!

Since you are asked about a "configuration" (that is, problems involving geometric shapes) you may try starting by drawing the shape and thinking about what common things you know about that shape.

Answer 4

For example:

Answer 5

Then look at the information the problem give you:
"a rectangular is 42 feet longer than it is wide. its perimeter is 276 feet. "
How can you use this information?
1.) How do you interpret the phrase " 42 feet longer than it is wide" in terms of L and W?

Answer 6

2l+2w=276?

Answer 7

Yes!!! That's the interpretation of the second piece of information!

Answer 8

The first sentence: 42 feet longer than it is wide
This can be one of two things:
L = W + 42 or W = L + 42

Answer 9

so either one would be correct?

Answer 10

Excellent! In most problems like these the math will work itself out.

Answer 11

So you now have two equations for your system: (part a done)
2L + 2W = 276
L = W + 42

Answer 12

but do i get to the solution?

Answer 13

That's the second part of your question. Using "substitution".

Answer 14

See how the second equation is already solved for L...?

Answer 15

yes

Answer 16

You can "substitute" (W + 42) in for "L" in the first equation to get:

2(W + 42) + 2W = 276

then solve for W.

Answer 17

this is where i get lost and or confused

Answer 18

No problem. Do you recall a technique called "distribution" as a way of "getting rid of parentheses?

Answer 19

yes

Answer 20

To solve, start by distributing:
2(W + 42) + 2W = 276
2W + 84 + 2W = 276

notice: after distribution I don't include the ( )

Answer 21

does that make sense ...?

Answer 22

where did 84 come from?

Answer 23

yes it is starting to make sense

Answer 24

good question. 84 is 2 times 42. Distribute is just fancy for multiplying.
2(W + 42) ---> 2 * W + 2 * 42 ---> 2W + 84

Answer 25

got it ;-)

Answer 26

Great!

Answer 27

So...
2W + 84 + 2W = 276

What do you do now?

Answer 28

divide 2w on both sides?

Answer 29

You could.... but that would get messy... Remember, when you divide both sides by anything you have to divide everything!!
\[\frac{ 2W + 84 + 2W }{ 2W } = \frac{ 276 }{ 2W}\]
which is the same as
\[\frac{ 2W}{2W} + \frac{84}{2W} + \frac{2W }{ 2W } = \frac{ 276 }{ 2W}\]

Hint: htere is a better way to proceed.

Answer 30

May I suggest "gathering like terms" :)

Answer 31

that sounds alot easier :-)

Answer 32

I'm glad you think so! Usually, you should save division as the last step.
So...
2W + 84 + 2W = 276 (combine the W's)
4W + 84 = 276

Answer 33

there are so many steps to this

Answer 34

It may seem that way, but we have only really taken three steps ...

2(W + 42) + 2W = 276 (first step was substitution)
2W + 84 + 2W = 276 (second step was distribution)
4W + 84 = 276 (third step was combining like term)
(there are three more to finish this problem)

You do much more complicated things everyday - for example, using a computer to talk about math online !)

Answer 35

And when you work a bunch of these problems, the steps start making more sense so you don't second guess what step comes next.

Answer 36

true that. so what is the next step?

Answer 37

Let's get the term with the W in it by itself, that is, you want to have the following:
4W = (a number)
but you have 4W + 84 = 276. How do you "move" or "cancel" the +84 ?

Answer 38

divide 4w on both sides?

Answer 39

If we had 4W = #, then dividing by 4 would be a great step. But we are not there yet,,,,

Answer 40

What happens if we subtract 84 from both sides:

4W + 84 - 84 = 276 - 84

Answer 41

276-84=192

Answer 42

good. And what about the left side?

Answer 43

4W + 84 - 84 would simplify how?

Answer 44

4w+0?

Answer 45

Yes... Is that the same as 4W (without the +0)

Answer 46

yes?

Answer 47

Yes!

Answer 48

Example: 2 + 0 is just 2.

Answer 49

yes

Answer 50

Almost there....
2(W + 42) + 2W = 276 (first step was substitution)
2W + 84 + 2W = 276 (second step was distribution)
4W + 84 = 276 (third step was combining like term)
4W = 192 (4th step was to subtract 84 from both sides)

and the final step to solve for W is ... your favorite!!! divide both sides of the equation by 4

Answer 51

lol

Answer 52

i object to that remark lol

Answer 53

jk

Answer 54

:)

Answer 55

192 divided 4 is 48

Answer 56

Lovely! So now you know the width is 48.

Look back on the original problem and the equations you created in part 1 of this problem. How can you find the length?

Answer 57

divide 48 into 276?

Answer 58

You are kinda obsessed with dividing things!

Answer 59

sorry

Answer 60

The original problem states that the length is 42 feet longer that the width.

Answer 61

thats the only way i have been used to doing things aside from multiplying things

Answer 62

We'll get to that in a sec...

Answer 63

If you know that the width is 48 feet, and that the length is 42 feet longer than that.. then you can say that the length is ....
48 feet + 42 feet ...

Answer 64

90?

Answer 65

Why the ? You got it!

Answer 66

no i wasnt sure if i was supposed to actually add them together

Answer 67

sure, so long as the numbers have the same units. For example, If you have 2 apples and you get another 5 apples, then you have 7 apples total. 2 apples + 5 apples = 7 apples

Answer 68

you are making me hungry lol.jk

Answer 69

I know, right!

Answer 70

Look... What you just had to solve was called a "multi-step linear equation" (I think you can guess why it's a multi-step). I would strongly suggest that you practice some "one-step" and "two-step" problems to make sure you master this "dividing both sides" "subtract from both sides" types of thing before continuing with problems like this. You would feel less confused, and more confident.

Take a look at Khan Academy: videos and practice
https://www.khanacademy.org/math/algebra-basics/core-algebra-linear-equations-inequalities

Answer 71

i definitely need practice

Answer 72

so whats the final step to this looooong process?

Answer 73

Just to state the answer in terms of the problem:

The rectangle is 48 feet by 90 feet.

Answer 74

that would be the solution?

Answer 75

Well there are two parts to the question:

1.) Create the system of equations
answer was the two equations:
2L + 2W = 276
L = W + 42

2.) Use substitution to solve
This was the long part:
2(W + 42) + 2W = 276 (first step was substitution)
2W + 84 + 2W = 276 (second step was distribution)
4W + 84 = 276 (third step was combining like term)
4W = 192 (4th step was to subtract 84 from both sides)
W = 48 (5th step was to divide both sides by 4)

Since W = 48, then substituting 48 in for W in the 2nd equation from part one you get:
L = W + 42
L = 48 + 42
L = 90

So the final result is that the rectangle is 48 feet wide and 90 feet long.