What is the value of x, 2x/3 + y/ 2 = 4 and x/3 - y/2 = 1?
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Understanding the Problem: Finding the Value of x in a System of Equations
The question asks us to find the value of the variable 'x' in a given system of two linear equations. A system of equations is a set of two or more equations that contain the same variables. The solution to a system is the set of values for the variables that satisfy all equations in the system simultaneously.
The System of Equations Given
We are provided with the following two equations:
- \(\frac{2x}{3} + \frac{y}{2} = 4\)
- \(\frac{x}{3} - \frac{y}{2} = 1\)
Solving the System of Equations: Using the Elimination Method
To find the value of x, we can use a method called elimination. The elimination method works well here because the terms involving 'y' in both equations (\(+\frac{y}{2}\) and \(-\frac{y}{2}\)) have the same coefficient but opposite signs. This means that if we add the two equations together, the 'y' terms will cancel out, or be "eliminated".
Step-by-Step Calculation to Find x
Let's add Equation 1 and Equation 2:
Equation 1: \(\frac{2x}{3} + \frac{y}{2} = 4\)
Equation 2: \(\frac{x}{3} - \frac{y}{2} = 1\)
Adding the corresponding sides of the equations:
\((\frac{2x}{3} + \frac{y}{2}) + (\frac{x}{3} - \frac{y}{2}) = 4 + 1\)
Group the 'x' terms and 'y' terms:
\((\frac{2x}{3} + \frac{x}{3}) + (\frac{y}{2} - \frac{y}{2}) = 5\)
Combine the terms. The 'y' terms cancel out:
\(\frac{2x + x}{3} + 0 = 5\)
\(\frac{3x}{3} = 5\)
Simplify the equation:
\(x = 5\)
Result: The Value of x
By applying the elimination method to the given system of linear equations, we successfully isolated the variable x and found its value.
The value of x is 5.
Revision Table: Steps for Solving Systems by Elimination
| Step Number | Action | How it Applies Here |
|---|---|---|
| 1 | Identify the system of equations. | Two equations with x and y are given. |
| 2 | Look for variables with coefficients that are the same or opposites. | The coefficients for the 'y' term are \(\frac{1}{2}\) and \(-\frac{1}{2}\) (opposites). |
| 3 | Add or subtract the equations to eliminate one variable. | Adding the equations eliminates the 'y' variable. |
| 4 | Solve the resulting equation for the remaining variable. | We got \(\frac{3x}{3} = 5\), which simplifies to \(x = 5\). |
| 5 | (Optional but Recommended) Substitute the value back into one original equation to find the other variable. | Substitute x=5 into \(\frac{x}{3} - \frac{y}{2} = 1\) to find y. |
| 6 | (Optional) Check the solution in both original equations. | Verify that the found x and y values satisfy both equations. |
Additional Information: Different Methods for Solving Systems
While the elimination method was very efficient for this particular system, there are other common methods used to solve systems of linear equations:
- Substitution Method: Involves solving one equation for one variable (like solving for x in terms of y), and then substituting that expression into the other equation.
- Graphing Method: Involves plotting both equations on a coordinate plane. The point where the lines intersect is the solution to the system. This method can sometimes be less precise if the intersection point is not exactly on grid lines.
- Matrix Method: More advanced methods using matrices can also be used, especially for larger systems with many variables.
Each method has its advantages depending on the form of the equations in the system.
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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?
The sum of two numbers is 66 and their difference is 22. What is the ratio of the two numbers?
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?
