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# linear systems and problem solving

you designed one row in your garden to broccoli and pea plants. Each broccoli plant needs 12 inches of space and each pea plant needs 6 inches of space. The row is 10 feet (120 inches) long. If you want a total of 13 plants, how many of each plant can you have.

The above three answers agree and are correct, but I find all of them excessively wordy - students get confused by an excess of teacher words... so, assuming they understand the basics and can get this system of linears:

12b + 6p = 120

b + p     = 13,

I'd multiply the latter eqaution by 6, on both sides, and subtract it from the first one:

12b + 6p  = 120

-(6b + 6p) = 13*6 = 78

-------------------------------

12b  - 6b  = 120 - 78 = 42

so, 6b = 42   ==>  b = 7, and we are nearly done: p = 13 - 7 = 6

WHY do it this way? It eliminates p to solve for b but you easily get the value of p too.

****First part of the problem:    Each broccoli plant needs 12 inches of space and each pea plant needs 6 inches of space. The row is 10 feet (120 inches) long.

You can use whatever letter you wish but I like to try to match the letters so I can easily determine what I am dealing with. For this example, we will use B=broccoli and P=Pea.

12(B) + 6(P) = 120      This equestion translates what the first problem states. Since all the units of measure are already in inches, we do not need to convert anything. So it states that broccoli needs 12 inches [hence the 12(B)] we use the letter 'B' since we do not know the number of broccoli plants. It also states that the pea plants takes 6 inches [hence 6(P)] we use the letter 'P' since we do not know the number of pea plants. Both added together will 120 inches.

****Second part of the problem:     If you want a total of 13 plants

So this tells me when I add the number of broccoli (B) plus the number of pea (P) plants, I should have a total of 13 hence B+P=13.

I have two equations:    12(B) + 6(P) = 120

B+P=13

Decide which one you wish to solve first which can be either Broccoli or Pea. We will solve for Broccoli (B) so in the equation B+P=13, we will subtract 'P' from both sides.

B + P = 13

-P         -P

B = 13 - P     This is what we are left with after subtracting 'P' from both sides. Now in the other equation we will substitute this equation into the 'B' which will look like this:

12(B) + 6(P) = 120                  B = 13 - P

[12(13-P)] +6P=120      Now we just multiply it out. I inserted brackets just so u know that the first part is seperate from the 6p. so we multiply 12*13 and 12*(-P) and we get the following:

156-12P+6P=120     We will now add the like terms together so the -12P+6P = -6P which gives us:

156-6P=120        Since we want to get the letter on one side, we will subtract 156 from both sides

so we will have -6P=120-156 which will leave us with:

-6P=-36           We divide by -6 (negative six) on both sides which will leave 'P' by itself

P=-36/-6        36 divided by 6 =6. since both numbers are negative we get a positive number

So P=6    Now to check our work, we will plug this number into the total number of plants equation

B=13-P          B=13-6          B=7       So B=7 & P=6

So now lets plug in these numbers into the other equation:    12(B) + 6(P) =120

12(B) + 6(P) = 120       12(7) + 6(6) =120       84+36=120        120=120

B+P=13           7 + 6 =13          13=13

Our figures are correct since both equations work out with our solution.

Since each broccoli plant takes up 12 inches and each pea plant takes up 6 inches and you have a total of 120 inches, you know that (12 inches*the number of broccoli plants) + (6 inches*the number of pea plants) will equal 120 inches.  12b + 6p = 120.  Then you also know that the number of broccoli and pea plants added together is 13 since you want a total of 13 plants.  b + p = 13.  If you take this last equation and solve for b you will get b = 13 - p.  Then look at your first equation and substitute the b for 13 - p:  12(13-p) + 6p = 120.  Simplify by distributing the 12*13 and 12*(-p) and get 156 - 12p + 6p = 120.  Combine like terms:  156 - 6p = 120.  Subtract 156 from both sides of the equation:  -6p = -36.  Divide by -6:  p = 6.  Substitute 6 in for p into either equation.  The second one is simplest:  b + 6 = 13.  Subtract 6 from both sides:  b = 7.  7 broccoli plants and 6 pea pods

We will begin by defining our variables.  I will use the following varaibles, though you can choose others:

x = number of broccoli plants

y = number of pea plants

The first equation will be the number of broccoli plants plus the number of pea plants equaling the number of total plants.

x  + y = 13

The second equation will relate the amount of space needed for the broccoli plants (12x) plus the amount of space needed for the pea plants (6y) equaling the amount of space used by all of the plants (120).

12x + 6y = 120

I am going to use substitution method, but you can use elimination if you prefer:

x + y = 13                      Original equation

x = 13 - y                       Subtract y from each side

12x + 6y = 120                Original equation

12(13 - y) + 6y =120        Substitution

156 - 12y + 6y = 120        Distribute the 12

156 - 6y = 120                 Simplify

-6y = 120 -156                 Subtract 156 from each side

-6y = -36                         Simplify

y = -36/(-6)                     Divide both sides by - 6

y = 6                               Simplify

Using the equation you found earlier:

x = 13 -y

x = 13 - 6                         Substitute

x = 7                               Simplify

You can plant 7 broccoli plants and 6 pea plants.

I like to start by figuring out what is being asked first.

• How many of each plant can you have?

Then I like to know what information is being given

• We want 13 plants total
• Broccoli needs 12 in (B)
• Peas need 6 in (P)
• The row is 10 ft. (120 in) long

So the equations are

P + B = 136P + 12B = 120 (keeping in mind that the units are inches)

So we can solve for P in the equation

P + B = 13

-B    -B

P       = 13 – B

Now you substitute in P = 13 – B in the other equation,

6(13 – B) + 12B = 120

And now solve for B

78 – 6B + 12B = 120

78 + 6B = 120

-78           -78

6B = 42

B = 7

Now P + B =13

P + 7 = 13

-7     -7

P = 6

So the answer is 6 Pea Plants and 7 Broccoli Plants