Solving a System of Linear Equations: Finding the Value of x
When dealing with systems of linear equations, it's essential to understand the methods available for solving such systems. In this article, we will explore the solution of the following system of linear equations:
System of Linear Equations
Consider the system given by:
3x - 2y 0 xy 3The problem statement specifically asks for the value of x in the solution set, rounded to the nearest tenth. To achieve this, we will use the elimination method for solving the system of equations.
The Elimination Method
The elimination method involves manipulating the equations to eliminate one of the variables, thereby simplifying the system. Here, we will focus on finding the value of x.
Step 1: Multiply the Second Equation
Let's start with the second equation, (xy 3). We will multiply both sides of this equation by 2 to set it up for elimination with the first equation:
New second equation: [2xy 6]
Step 2: Add the Equations
Now, we can add the modified second equation to the first equation. This process eliminates the variable y:
First equation: [3x - 2y 0]
New second equation: [2xy 2y 6]
Combined equation: [(3x - 2y) (2xy 2y) 0 6]
This simplifies to:
[3x 2xy 6]
Notice that the y terms cancel out:
[3x 2xy - 2y 0 6]
[3x 0 6]
Thus, we have the simplified equation:
[3x 6]
Step 3: Solve for x
To solve for x, we divide the entire equation by 3:
[x dfrac{6}{3} 2]
However, the problem asks us to round the answer to the nearest tenth. Since 2 is already a whole number, it remains 2.0 when rounded to the nearest tenth.
Revisiting the Example
The question specifically mentions solving for x in the context of the first equation, (3x - 2y 0). Given the solution of the system, we have (x 2). But if we need to verify using the second equation and round appropriately:
The second equation is (xy 3). Substituting (x 2), we get:
[2y 3]
[y dfrac{3}{2} 1.5]
Lets confirm the initial setup:
[3(2) - 2(1.5) 6 - 3 0]
[2(1.5) 3]
Both equations are satisfied. Now, let's carefully re-examine the specified process for the solution:
Revisiting the Specified Process
Given the original equations:
3x - 2y 0 xy 3A correct approach would be to solve these using the given guidelines:
New second equation: [2xy 6]
Add to the first equation:
[3x - 2y 2xy 2y 0 6]
[3x 2xy 6]
Rearrange to isolate x:
[3x 6 - 2xy]
[3x 6]
[x dfrac{6}{5} 1.2]
Thus, the value of x, rounded to the nearest tenth, is 1.2.
Conclusion
Using the elimination method, we found that the value of x is 1.2. This resolves the system of equations and answers the problem as required. The key is careful manipulation and following the rules of algebra to solve for the desired variable.