Sorry this is a really stupid question.
Ive been going back to basics and im a bit lost (probably tired so I cant see the obvious)
so im working out 87v+38w=d
ive done the basics pretty well and have d=1.
Now backwards substitution is spinning my head a bit.
11=87-2*38
5=38-3*11
1=11-2*5
So using backwards substitution I have:
1=11-2(38-3*11)
1=7*11-2*38
This is where im a bit unsure about the order so we have reduced the equation down and im not entirely sure I understand intuitively where the 7 came from.
have we (11-(-3))/2=7
Again sorry this is such a basic question but I want to be sure I understand what is happening explicitly.
No need to apologize! It's great that you're seeking clarification. Let's break down the steps and understand what's happening explicitly.
Starting with your equation:
1 = 7*11 - 2*38
To arrive at this equation using backward substitution, you substitute the values you found in the previous steps:
1 = 11 - 2*(38 - 3*11) [substituting 38 - 3*11 = 5 from the previous step]
1 = 11 - 2*5
1 = 11 - 10
So far, everything is straightforward. Now, we need to simplify the expression further:
1 = 1
The "7" you mentioned does not appear in this equation, so the expression is already simplified to its simplest form. Therefore, there is no need to divide anything or involve "-3" in the process.
In summary, when using backward substitution, you substitute the values you found in the previous steps back into the original equation. In this case, the original equation was simplified to "1 = 1," so no additional simplification is required.
No worries, there's no such thing as a stupid question! Let's break it down and understand the process of backwards substitution in this equation.
First, you correctly found the values for d=1 using the steps you provided. Now, in order to solve for v and w, we need to work backward from the bottom equation (1=7*11-2*38) to the top equation (87v+38w=d).
To start, let's review the three equations you derived using backwards substitution:
1=11-2(38-3*11)
1=7*11-2*38
From the first equation, we can see that 11-2(38-3*11) simplifies to 11-2(38-33), which further simplifies to 11-2(5).
Now, the key to understanding where the 7 comes from is to evaluate the expression 5/2 differently. Instead of thinking about division, let's think about it in terms of multiples of 2. We need a multiple of 2 that gets us as close to 5 as possible. The closest multiple of 2 to 5 is 4 (2 * 2 = 4).
So, we can write 5 as 4 + 1. Now, let's substitute this into the equation:
11 - 2(5) = 11 - 2(4 + 1)
Distribute the -2:
11 - 2(4) - 2(1)
Simplify:
11 - 8 - 2
Combine like terms:
1
Therefore, the first equation simplifies to 1.
Moving on to the second equation, we have 1=7*11-2*38.
Again, let's focus on the term 38/2. Instead of thinking about it as division, let's think about it as multiples of 2. In this case, the closest multiple of 2 to 38 is 36 (2 * 18 = 36).
So, we can write 38 as 36 + 2. Now, let's substitute this into the equation:
1 = 7 * 11 - 2 * (36 + 2)
Distribute the -2:
1 = 7 * 11 - 2 * 36 - 2 * 2
Simplify:
1 = 7 * 11 - 72 - 4
Combine like terms:
1 = 77 - 76
Solve:
1 = 1
Therefore, the second equation simplifies to 1.
So, by using backwards substitution, we found that v and w are both equal to 1 when d=1. The 7 was derived from expressing the 5 as 4 + 1 and substituting that into the equation.
I hope this explanation clarifies the steps involved in backwards substitution. If you have any further questions, feel free to ask!