Simultaneous equations substitution method

In summary, the conversation discusses a problem involving finding the weights of two objects using the substitution method. One person is stuck and shares their thought process while the other person suggests alternative equations to use. Eventually, the problem is solved by substituting one equation into the other and solving for the unknown variables. The conversation also notes that there are multiple methods for solving such equations.
  • #1
hackedagainanda
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Homework Statement
A) Use the clue and the diagram to write a pair of simultaneous equations.
B) Solve the equations to find x and y
C) Check your solution to see if it makes the seesaw balance

Problem 16. The sum of the weights is 24.
Relevant Equations
Fulcrum equation Ax = By where A and B are the respective weights and x and y are the distant between them needed for balancing
20200915_214309.jpg
I'm really stuck on this one, I was able to get the answer but not by the substitution method.

So its the weight as A and B so I get A + B = 24
A(3) = B(5) so in my head I calculate a few pairs, 3 x 5 = 15 but 3 + 5 only = 8 so the next pair would be 10 and 6 which is still to small so I move on to 15 and 9 which does = 24 and solves the equation but I failed to use the substitution method which to me means I haven't really solve the problem.
 
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  • #2
So, your unknown variables are the two weights, x and y (or A and B, as you said). You've already told us that x + y = 24, that is one of the two equations that you need to solve this. Can you think of another relevant equation? It would represent another requirement on the distribution of weight. For example x = 0 doesn't work, y = 0 doesn't work, why not?
 
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  • #3
There has to be some weight on both sides for it to balance so neither x or y can be 0.

I think I got it. So I try 3x = 5y which when gives me x = 1 and (2/3) y so
x+ y is (8/3) = 24 which gives me 9, then its x + 9 = 24 subtract 9 from both sides and get x = 15. Thanks for the help!
 
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  • #4
Yes! good work.

I'm not sure I followed your description of the process though. Here's one way I would do it:
1) x + y = 24 => y = 24 - x
2) 3x =5y => x = (5/3)y
substitute 2) into 1) => y = 24 - (5/3)y then solve for y, y + (5/3)y = 24, y = 24/(1 + 5/3), y = 9
then find x with one of the first two equations, x = 24 - 9 = 15.

BTW, there are a lot of different but equally good way of solving these equations.
 
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What is the simultaneous equations substitution method?

The simultaneous equations substitution method is a mathematical technique used to solve a system of two equations with two unknown variables. It involves solving one of the equations for one of the variables and then substituting that expression into the other equation to solve for the remaining variable.

When is the simultaneous equations substitution method used?

This method is used when there are two equations with two unknown variables and the equations are not easily solvable by other methods such as elimination or graphing. It is also useful when one of the equations involves a variable with a coefficient of 1.

What are the steps for solving a system of equations using the substitution method?

The steps for solving a system of equations using the substitution method are as follows:

  1. Choose one of the equations and solve it for one of the variables.
  2. Substitute the expression for the solved variable into the other equation.
  3. Solve the resulting equation for the remaining variable.
  4. Substitute the value found for the remaining variable back into the original equation to find the value of the first variable.
  5. Check the solution by plugging in the values for both variables into the other equation and ensuring that it is satisfied.

What are the advantages of using the simultaneous equations substitution method?

One advantage of this method is that it can be used to solve systems of equations with two unknown variables that cannot be easily solved by other methods. It also allows for a systematic approach to finding the solution, making it less prone to errors compared to other methods.

What are the limitations of the simultaneous equations substitution method?

This method can only be used for systems of equations with two unknown variables. It also requires one of the equations to have a variable with a coefficient of 1, which may not always be the case. Additionally, if the equations are not linear, this method cannot be used and other techniques such as graphing or matrix operations may be needed.

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