Solving System Of Equations T = 5c + 35 And T = 10c Number Of Solutions
When faced with a system of equations, a fundamental question arises: how many solutions does this system possess? Delving into this question involves analyzing the equations themselves, understanding their relationships, and employing various techniques to determine the solution count. In this article, we will meticulously examine the given system of equations:
t = 5c + 35
t = 10c
and unravel the mystery of its solution count. We will not only pinpoint the number of solutions but also dissect the reasoning behind our determination. This exploration will provide a comprehensive understanding of systems of equations and their solution characteristics.
Unveiling the Number of Solutions
To ascertain the number of solutions this system holds, we can employ a variety of methods, such as substitution, elimination, or graphical analysis. Let's opt for the substitution method, which involves solving one equation for one variable and substituting that expression into the other equation. This approach allows us to reduce the system to a single equation with a single variable, making it easier to solve.
From the second equation, we readily have t = 10c. Now, we substitute this expression for t into the first equation:
10c = 5c + 35
This substitution yields an equation solely in terms of c. Solving for c, we subtract 5c from both sides:
5c = 35
Dividing both sides by 5, we obtain:
c = 7
Now that we have the value of c, we can substitute it back into either of the original equations to find the value of t. Let's use the second equation, t = 10c:
t = 10 * 7
t = 70
Therefore, we have found a unique solution: c = 7 and t = 70. This indicates that the system of equations has exactly one solution. Graphically, this corresponds to the two lines intersecting at a single point, which represents the unique solution (7, 70).
The Rationale Behind the Single Solution
The determination that this system possesses one unique solution stems from the nature of the equations themselves. We have two linear equations in two variables. Each equation represents a straight line when graphed on a coordinate plane. The solution to the system corresponds to the point(s) where these lines intersect. There are three possible scenarios:
- The lines intersect at one point: This signifies a unique solution, as we found in our case.
- The lines are parallel and do not intersect: This implies that there are no solutions, as the equations are inconsistent.
- The lines are coincident (they are the same line): This indicates infinitely many solutions, as every point on the line satisfies both equations.
In our case, the lines represented by the equations t = 5c + 35 and t = 10c have different slopes (5 and 10, respectively) and different y-intercepts (35 and 0, respectively). This guarantees that the lines are neither parallel nor coincident. Therefore, they must intersect at exactly one point, leading to a unique solution.
Let's delve deeper into why different slopes are crucial for a unique solution. The slope of a line dictates its steepness and direction. If two lines have the same slope, they are either parallel (if they have different y-intercepts) or coincident (if they have the same y-intercept). Parallel lines never intersect, while coincident lines intersect at every point. Only lines with differing slopes will intersect at a single, distinct point.
In our system, the slopes of 5 and 10 clearly distinguish the lines, ensuring their intersection. The different y-intercepts (35 and 0) further confirm that they are not coincident. This combination of distinct slopes and y-intercepts is the cornerstone of our conclusion that the system has one and only one solution.
Exploring Alternative Solution Methods
While we employed the substitution method to solve the system, it's worthwhile to explore alternative approaches to further solidify our understanding and demonstrate the consistency of the solution.
Elimination Method
The elimination method involves manipulating the equations to eliminate one variable, thereby reducing the system to a single equation with a single variable. In our case, we can subtract the first equation from the second equation to eliminate t:
(t = 10c) - (t = 5c + 35)
This subtraction yields:
0 = 5c - 35
Adding 35 to both sides, we get:
5c = 35
Dividing both sides by 5, we again obtain:
c = 7
Substituting this value of c back into either original equation, we find t = 70, confirming our previous solution.
The elimination method provides an alternative pathway to the same solution, reinforcing the accuracy of our result and highlighting the flexibility in solving systems of equations.
Graphical Method
The graphical method involves plotting the equations on a coordinate plane and visually identifying the point(s) of intersection. The graph of t = 5c + 35 is a straight line with a slope of 5 and a y-intercept of 35. The graph of t = 10c is a straight line with a slope of 10 and a y-intercept of 0.
When these lines are plotted, it becomes evident that they intersect at a single point, which corresponds to the solution (c, t) = (7, 70). This visual representation provides an intuitive understanding of the solution and corroborates our findings from the algebraic methods.
The graphical method offers a valuable visual confirmation of the solution, especially for linear systems. It allows us to see the intersection of the lines, providing a clear picture of the solution's existence and uniqueness.
The Significance of One Unique Solution
The fact that this system of equations has one unique solution carries significant implications. It means that there is one and only one pair of values for c and t that simultaneously satisfies both equations. This has practical relevance in various contexts where systems of equations are used to model real-world phenomena.
For instance, if c represents the number of items produced and t represents the total cost, the unique solution would indicate the specific production level and corresponding cost where the two cost equations intersect. This information can be crucial for decision-making in business and economics.
The uniqueness of the solution also implies that the system is well-defined and consistent. There are no conflicting constraints or redundancies in the equations. This is essential for accurate modeling and prediction.
In summary, the existence of a single unique solution to a system of equations signifies a well-behaved system with a specific and meaningful solution that can be applied in various real-world scenarios.
Conclusion
In this comprehensive exploration, we meticulously examined the system of equations:
t = 5c + 35
t = 10c
and definitively determined that it has exactly one solution. We arrived at this conclusion through algebraic methods, including substitution and elimination, and further validated our findings with graphical analysis. The rationale behind the single solution lies in the distinct slopes and y-intercepts of the lines represented by the equations, ensuring their intersection at a single point.
The significance of a unique solution extends beyond the mathematical realm, as it indicates a well-defined and consistent system with practical applications in various fields. This understanding of solution counts and their underlying principles is fundamental to mastering systems of equations and their role in problem-solving.
This detailed examination provides a thorough understanding of the solution count for the given system of equations. By employing various methods and delving into the underlying principles, we have not only found the solution but also gained valuable insights into the nature of systems of equations and their applications. Understanding these concepts is crucial for anyone working with mathematical models and real-world problems.
System of equations, solutions, substitution method, elimination method, graphical method, slopes, y-intercepts, unique solution, consistent system, linear equations, intersection point.
Solving Systems of Equations How Many Solutions Does This System Have?
