What is the value of u in the system of equations 3 (2u + v) = 7uv, 3 (u + 3v) = 11uv?
1
Solving System of Equations: Finding the Value of u
We are given a system of two equations with two variables, u and v:
- \(3 (2u + v) = 7uv\)
- \(3 (u + 3v) = 11uv\)
These are non-linear equations because of the \(uv\) term. To solve them, we can try to transform them into a simpler form, possibly a system of linear equations. A common technique when variables appear in a product like this is to divide by the product of the variables, assuming the variables are not zero.
Transforming the System of Equations
Let's assume \(u \neq 0\) and \(v \neq 0\). We can divide both sides of each equation by \(uv\).
Equation 1 Transformation:
Start with the first equation:
\[3 (2u + v) = 7uv\]Distribute the 3 on the left side:
\[6u + 3v = 7uv\]Divide both sides by \(uv\):
\[\frac{6u + 3v}{uv} = \frac{7uv}{uv}\] \[\frac{6u}{uv} + \frac{3v}{uv} = 7\]Cancel out common terms:
\[\frac{6}{v} + \frac{3}{u} = 7\]Equation 2 Transformation:
Now, transform the second equation:
\[3 (u + 3v) = 11uv\]Distribute the 3 on the left side:
\[3u + 9v = 11uv\]Divide both sides by \(uv\):
\[\frac{3u + 9v}{uv} = \frac{11uv}{uv}\] \[\frac{3u}{uv} + \frac{9v}{uv} = 11\]Cancel out common terms:
\[\frac{3}{v} + \frac{9}{u} = 11\]Forming a Linear System
We now have a new system of equations in terms of \(1/u\) and \(1/v\):
- \(\frac{3}{u} + \frac{6}{v} = 7\)
- \(\frac{9}{u} + \frac{3}{v} = 11\)
Let \(x = \frac{1}{u}\) and \(y = \frac{1}{v}\). The system becomes a linear system in variables \(x\) and \(y\):
- \(3x + 6y = 7\)
- \(9x + 3y = 11\)
We can solve this linear system using methods like substitution or elimination. Let's use the elimination method.
Solving the Linear System using Elimination
We want to eliminate one of the variables, say \(y\). We can multiply the second equation (\(9x + 3y = 11\)) by 2 so that the coefficient of \(y\) becomes 6, matching the first equation:
\[2 \times (9x + 3y) = 2 \times 11\] \[18x + 6y = 22\]Let's call this new equation Equation 3:
- \(18x + 6y = 22\)
Now we subtract the first equation (\(3x + 6y = 7\)) from Equation 3:
\[(18x + 6y) - (3x + 6y) = 22 - 7\] \[18x + 6y - 3x - 6y = 15\] \[15x = 15\]Now, solve for \(x\):
\[x = \frac{15}{15}\] \[x = 1\]Finding the Value of u
We defined \(x = \frac{1}{u}\). We found that \(x = 1\). So:
\[\frac{1}{u} = 1\]Solving for \(u\):
\[u = \frac{1}{1}\] \[u = 1\]We can also find \(y\) and consequently \(v\) by substituting \(x=1\) into either of the linear equations. Using \(3x + 6y = 7\):
\[3(1) + 6y = 7\] \[3 + 6y = 7\] \[6y = 7 - 3\] \[6y = 4\] \[y = \frac{4}{6}\] \[y = \frac{2}{3}\]Since \(y = \frac{1}{v}\), we have:
\[\frac{1}{v} = \frac{2}{3}\] \[v = \frac{3}{2}\]So, the solution we found is \(u=1\) and \(v=3/2\).
Note: The division by \(uv\) assumed \(u \neq 0\) and \(v \neq 0\). If \(u=0\), the original equations become \(3(v) = 0 \implies v=0\) and \(3(3v)=0 \implies 9v=0 \implies v=0\). So, \((0,0)\) is also a solution to the original system. However, the options provided for the value of u are non-zero, indicating the question seeks the non-zero solution.
Conclusion on the Value of u
Based on the method of transforming the system by dividing by \(uv\), we found a non-zero solution where the value of u is 1.
The value of u is 1.
| Original Equation | Transformed Equation (\(x=1/u, y=1/v\)) |
|---|---|
| \(3(2u+v) = 7uv\) | \(3x + 6y = 7\) |
| \(3(u+3v) = 11uv\) | \(9x + 3y = 11\) |
Revision Table: Key Steps to Solve the System
| Step | Description | Detail |
|---|---|---|
| 1 | Analyze the system | Identify as non-linear due to the \(uv\) term. |
| 2 | Transform equations | Assuming \(u,v \neq 0\), divide by \(uv\) to get equations in terms of \(1/u\) and \(1/v\). |
| 3 | Substitute variables | Let \(x = 1/u\) and \(y = 1/v\) to form a linear system. |
| 4 | Solve linear system | Use elimination or substitution to find values for \(x\) and \(y\). |
| 5 | Convert back | Use \(u = 1/x\) and \(v = 1/y\) to find the original variables. |
Additional Information on Solving Non-Linear Systems
Solving systems of non-linear equations can be more complex than linear systems. Different techniques are used depending on the form of the equations.
- Substitution: If one equation can be easily solved for one variable in terms of the other, substitute this expression into the other equation.
- Elimination: Sometimes, equations can be added or subtracted to eliminate a variable, similar to linear systems, but this often requires transforming the equations first, as done in this problem.
- Graphical Methods: For simple systems, plotting the graphs of the equations can show intersection points which are the solutions. However, this is not always precise.
- Special Transformations: As seen here, specific forms (like terms being products of variables) might suggest specific transformations (like dividing by the product) to simplify the system.
It's important to check all potential solutions in the original equations, especially when transformations (like division) are used, which might exclude certain cases (like variables being zero).
Similar Questions
What is the solution of the equation x log 10 (10/3) + log 10 3 = log 10 (2 + 3 x) + x?
If a 3= 335 + b 3and a = 5 + b, then what is the value of a + b (given that a > 0 and b > 0)?
Five years ago, Ram was three times as old as Shyam. Four years from now, Ram will be only twice as old as Shyam. What is the present age of Ram?
The values of x and y from the equations x - y = 6 and x/3 + y/2 = 12 are:
What is the value of x, 2x/3 + y/ 2 = 4 and x/3 - y/2 = 1?
How many integral values of x and y satisfy the equation 5x + 9y = 7, where -500 < x < 500 and -500 < y < 500?
A person carries Rs. 500 and wants to buy apples and oranges out of it. If the cost of one apple is Rs. 5 and the cost of one orange is Rs. 7 then what is the number of ways in which a person can buy both apples and oranges using total amount?
How many pairs of natural numbers are there such that the difference of their squares is 35?
What is the maximum value of the expression \(\frac{1}{{{x^2}\; + \;5x\; + \;10}}?\)
A man who recently died left a sum of Rs. 3,90,000 to be divided among his wife, five sons and four daughters. He directed that each son should receive 3 times as much as each daughter receives and that each daughter should receive twice as much as their mother receives. What was the wife’s share?
Important Questions from Linear Equation in 2 Variable
If 2 x + 3 y = 17;
2 x+2 - 3 y+1 = 5
then the values of x and y are:
The solution of pair of linear equations \(\dfrac{1}{2}x+\dfrac{2}{3}y=-1,x-\dfrac{1}{3}y=3\) by the elimination method, is:
Kumar tried his skill at shooting at a fun fair. He has to hit the target and if he hits the target he gets 1 Rs. and if he misses he has to pay 50 paise. He attempted 25 shots and won 10 Rs. In how many did he hit the target?
Shyam spent half of his money and was left with as many as he had rupees before, but with half as many rupees as he had paise before. Which of the following is a possible amount of money he is left with?
Two bus tickets from city A to B and three tickets from city A to C cost Rs. 77, but three tickets from city A to B and two tickets from city A to C cost Rs. 73. What are the fares for cities B and C from A?
