Solving Systems Of Equations Determining The Solution For -3x + 4y = 12 And X/4 - Y/3 = 1
Determining the nature of solutions for a system of linear equations is a fundamental concept in algebra. In this article, we will delve into the system of equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1 to ascertain whether it possesses a unique solution and, if so, to identify that solution. We will meticulously explore the methods of solving such systems, including substitution, elimination, and graphical approaches. Furthermore, we'll discuss the conditions under which a system of linear equations has one solution, no solutions, or infinitely many solutions. Let's embark on this mathematical journey to unravel the solution to this intriguing problem.
Understanding Systems of Linear Equations
Before we dive into the specifics of the given equations, let's establish a solid foundation by understanding what systems of linear equations are and how they behave. A system of linear equations is a set of two or more linear equations involving the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. Geometrically, each linear equation represents a straight line, and the solution to the system corresponds to the point(s) where the lines intersect.
Types of Solutions
When dealing with systems of linear equations, three possible scenarios can arise regarding the nature of solutions:
- Unique Solution: The system has exactly one solution, meaning the lines intersect at a single point. This indicates that the equations are independent and consistent.
- No Solution: The system has no solution, implying the lines are parallel and never intersect. In this case, the equations are inconsistent.
- Infinitely Many Solutions: The system has infinitely many solutions, suggesting the lines coincide, meaning they are essentially the same line. The equations are dependent and consistent.
Methods for Solving Systems of Linear Equations
Several methods exist for solving systems of linear equations. Let's briefly discuss some of the most common approaches:
- Substitution Method: Solve one equation for one variable and substitute the expression into the other equation.
- Elimination Method: Multiply one or both equations by constants to make the coefficients of one variable opposites, then add the equations to eliminate that variable.
- Graphical Method: Graph both equations on the same coordinate plane and find the point(s) of intersection.
Now that we have a grasp of the fundamentals, let's apply these concepts to the specific system of equations presented in the problem.
Analyzing the Given Equations: -3x + 4y = 12 and (1/4)x - (1/3)y = 1
Our task is to determine whether the system of equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1 has a unique solution. If it does, we aim to identify the solution. To achieve this, we can employ either the substitution method or the elimination method. Let's opt for the elimination method, as it appears to be a more efficient approach in this scenario.
Transforming the Equations
Before we can apply the elimination method effectively, it's advantageous to eliminate the fractions in the second equation. To do so, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 12:
12 * [(1/4)x - (1/3)y] = 12 * 1
This simplifies to:
3x - 4y = 12
Now, our system of equations looks like this:
- -3x + 4y = 12
- 3x - 4y = 12
Applying the Elimination Method
Observe that the coefficients of 'x' in the two equations are opposites (-3 and 3). This makes the elimination method particularly convenient. We can add the two equations directly:
(-3x + 4y) + (3x - 4y) = 12 + 12
Simplifying, we get:
0 = 24
Interpreting the Result
The equation 0 = 24 is a contradiction, a mathematical impossibility. This outcome signifies that the system of equations is inconsistent and has no solution. The lines represented by these equations are parallel and never intersect.
Conclusion: No Unique Solution
Based on our analysis, the system of equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1 does not have a unique solution. The equations are inconsistent, and their graphical representation would be two parallel lines. Therefore, none of the provided options (A and B) are correct.
To further solidify your understanding, let's delve deeper into why this system has no solution and explore the characteristics of inconsistent systems of linear equations.
Deeper Dive: Inconsistent Systems and Parallel Lines
An inconsistent system of linear equations, as we've encountered in this problem, arises when the equations represent lines that are parallel. Parallel lines, by definition, never intersect, which translates to the system having no common solution. To gain a clearer picture, let's revisit the concept of the slope-intercept form of a linear equation.
Slope-Intercept Form
The slope-intercept form of a linear equation is given by:
y = mx + b
where:
- 'm' represents the slope of the line
- 'b' represents the y-intercept (the point where the line crosses the y-axis)
Parallel lines have the same slope but different y-intercepts. Let's rewrite our original equations in slope-intercept form to see this more clearly.
Converting to Slope-Intercept Form
-
-3x + 4y = 12 Add 3x to both sides: 4y = 3x + 12 Divide both sides by 4: y = (3/4)x + 3
-
(1/4)x - (1/3)y = 1 Multiply both sides by 12: 3x - 4y = 12 Subtract 3x from both sides: -4y = -3x + 12 Divide both sides by -4: y = (3/4)x - 3
Notice that both equations have the same slope (m = 3/4) but different y-intercepts (3 and -3). This confirms that the lines are parallel and the system has no solution.
Implications of No Solution
The absence of a solution in a system of linear equations has significant implications in various real-world applications. For example, in economics, if a system of equations representing supply and demand has no solution, it indicates that there is no equilibrium point where supply equals demand. In engineering, it might signify that a system of constraints cannot be satisfied simultaneously, requiring a redesign.
Beyond Two Equations: Systems with More Variables
The concepts we've discussed for two equations extend to systems with more variables. For instance, a system of three linear equations with three variables represents planes in three-dimensional space. The solution to the system corresponds to the point(s) where all three planes intersect. Similar to the two-variable case, a system with three variables can have a unique solution, no solution, or infinitely many solutions.
Methods for Solving Larger Systems
While substitution and elimination can be used for larger systems, more sophisticated techniques like Gaussian elimination and matrix methods become more practical. These methods provide a systematic approach to solving systems of linear equations, regardless of the number of variables.
Concluding Thoughts
Understanding the nature of solutions for systems of linear equations is crucial in various fields. By mastering the methods of solving these systems and interpreting the results, you'll be well-equipped to tackle a wide range of problems. In the specific case of the equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1, we've demonstrated that the system is inconsistent and has no solution, highlighting the importance of careful analysis and the connection between algebraic representations and geometric interpretations.
