Solving Linear Functions The Tables Represent In A System
In mathematics, a system of linear equations represents a set of two or more linear equations considered together. The solution to a system of linear equations is the point (or set of points) that satisfies all the equations in the system. This point represents the intersection of the lines when the equations are graphed. Solving systems of equations is a fundamental concept in algebra, with applications in various fields, including engineering, economics, and computer science. In this article, we will delve into how to solve a system of linear equations presented in table form. Specifically, we will analyze two tables representing two linear functions and determine their solution. By understanding the underlying principles and applying appropriate techniques, we can effectively find the point where these two lines intersect, thus solving the system.
Understanding Linear Functions and Systems of Equations
To effectively tackle the problem at hand, it's crucial to grasp the fundamentals of linear functions and systems of equations. A linear function is a function that forms a straight line when graphed. It can be represented in the form y = mx + b, where m represents the slope (the rate of change of y with respect to x) and b represents the y-intercept (the point where the line crosses the y-axis). Understanding this form is critical because it allows us to translate tabular data into algebraic equations, which are easier to manipulate and solve.
A system of linear equations, on the other hand, is a collection of two or more linear equations involving the same variables. The solution to such a system is the set of values for the variables that satisfy all equations simultaneously. Geometrically, this solution represents the point(s) where the lines corresponding to the equations intersect. There are several methods to solve systems of equations, including graphing, substitution, and elimination. Each method has its strengths and is suitable for different types of systems. For instance, the graphing method is visually intuitive but may not provide precise solutions, while algebraic methods like substitution and elimination offer more accuracy.
In the context of this problem, we are given two linear functions represented in tabular form. Our goal is to find the solution to the system formed by these two functions. This means we need to determine the point (x, y) that satisfies both sets of data. By converting the tabular data into linear equations and then applying appropriate solving techniques, we can find the solution. This process not only reinforces our understanding of linear functions and systems of equations but also highlights the practical application of these concepts in problem-solving.
Analyzing the First Linear Function
To begin, let's meticulously analyze the first table provided. This table represents a linear function, and our immediate goal is to determine the equation that corresponds to this function. Remember, a linear function can be expressed in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To find these values, we can use the data points provided in the table. The slope, m, represents the rate of change of y with respect to x, and can be calculated using any two points (x1, y1) and (x2, y2) from the table using the formula: m = (y2 - y1) / (x2 - x1).
By examining the table, we can choose two points to calculate the slope. For instance, let's select the points (-4, 26) and (-2, 18). Plugging these values into the slope formula, we get m = (18 - 26) / (-2 - (-4)) = -8 / 2 = -4. So, the slope of the first linear function is -4. This means that for every unit increase in x, y decreases by 4 units. Now that we have the slope, we need to find the y-intercept, b. The y-intercept is the value of y when x is 0. Looking at the table, we can see that when x is 0, y is 10. Therefore, the y-intercept, b, is 10.
With both the slope and the y-intercept determined, we can now write the equation for the first linear function. Substituting the values m = -4 and b = 10 into the slope-intercept form, y = mx + b, we get y = -4x + 10. This equation represents the linear relationship described by the first table. It's crucial to verify this equation by plugging in other points from the table to ensure it holds true. This step confirms the accuracy of our derived equation and sets the stage for solving the system of equations.
Analyzing the Second Linear Function
Now, let's shift our focus to the second table and apply a similar process to determine the equation of the second linear function. Just as we did with the first table, our aim is to express this function in the slope-intercept form, y = mx + b. This requires us to find the slope (m) and the y-intercept (b) using the data points provided.
To calculate the slope, m, we once again use the formula m = (y2 - y1) / (x2 - x1). Let's choose two points from the second table, for example, (-4, 14) and (-2, 8). Plugging these values into the formula, we get m = (8 - 14) / (-2 - (-4)) = -6 / 2 = -3. This indicates that the slope of the second linear function is -3. This means for every unit increase in x, y decreases by 3 units, which is a different rate of change compared to the first function.
Next, we need to find the y-intercept, b, which is the value of y when x is 0. By examining the second table, we can see that when x is 0, y is 2. Therefore, the y-intercept, b, for the second linear function is 2. Now that we have both the slope and the y-intercept, we can construct the equation for the second linear function. Substituting the values m = -3 and b = 2 into the slope-intercept form, y = mx + b, we obtain y = -3x + 2. This equation represents the linear relationship described by the second table. As with the first equation, it's important to verify this equation by plugging in other points from the table to ensure its accuracy. This step confirms that our equation correctly represents the data, setting us up to solve the system of equations.
Solving the System of Equations
With the equations for both linear functions determined, we are now ready to solve the system of equations. We have the following two equations:
- y = -4x + 10
- y = -3x + 2
To find the solution, which is the point (x, y) that satisfies both equations, we can use several methods. One effective method is the substitution method. Since both equations are already solved for y, we can set them equal to each other. This gives us:
-4x + 10 = -3x + 2
Now, we can solve for x. To isolate x, we can add 4x to both sides of the equation:
10 = x + 2
Next, we subtract 2 from both sides:
8 = x
So, the x-coordinate of the solution is 8. Now that we have the value of x, we can substitute it into either of the original equations to find the value of y. Let's use the second equation, y = -3x + 2:
y = -3(8) + 2 y = -24 + 2 y = -22
Therefore, the y-coordinate of the solution is -22. Thus, the solution to the system of equations is the point (8, -22). This point represents the intersection of the two lines represented by the equations. To ensure the accuracy of our solution, we can substitute both x = 8 and y = -22 into both original equations to verify that they hold true. This final check provides confidence in our result and completes the process of solving the system of equations.
Verifying the Solution
After finding a potential solution to a system of equations, it is crucial to verify its accuracy. This step ensures that the solution we have calculated indeed satisfies all the equations in the system. In our case, we found the solution to be (8, -22) for the system:
- y = -4x + 10
- y = -3x + 2
To verify, we will substitute x = 8 and y = -22 into both equations and check if the equations hold true.
For the first equation, y = -4x + 10, we substitute the values:
-22 = -4(8) + 10 -22 = -32 + 10 -22 = -22
The equation holds true, so the solution satisfies the first linear function.
Now, let's substitute the values into the second equation, y = -3x + 2:
-22 = -3(8) + 2 -22 = -24 + 2 -22 = -22
This equation also holds true, confirming that the solution satisfies the second linear function as well. Since the point (8, -22) satisfies both equations, we can confidently conclude that it is the correct solution to the system of equations. This verification step is a critical part of the problem-solving process, as it ensures that our answer is accurate and that we have correctly applied the concepts of linear functions and systems of equations.
Conclusion
In this comprehensive analysis, we have successfully navigated the process of solving a system of linear equations presented in table form. We began by understanding the fundamental concepts of linear functions and systems of equations, emphasizing the importance of the slope-intercept form (y = mx + b) in representing linear relationships. We then meticulously analyzed each table, extracting data points to determine the equations for the two linear functions. This involved calculating the slope (m) and y-intercept (b) for each function, which allowed us to express them in algebraic form.
Once we had the equations, y = -4x + 10 and y = -3x + 2, we employed the substitution method to find the solution to the system. By setting the two equations equal to each other and solving for x, we found the x-coordinate of the solution. We then substituted this value back into one of the original equations to find the y-coordinate. This process led us to the solution (8, -22), which represents the point of intersection of the two lines.
To ensure the accuracy of our solution, we performed a crucial verification step. We substituted the values x = 8 and y = -22 into both original equations and confirmed that they held true. This step reinforced our confidence in the solution and demonstrated a thorough understanding of the problem-solving process. Solving systems of equations is a fundamental skill in mathematics, with applications spanning various fields. By mastering this skill, we can tackle complex problems involving multiple variables and relationships, making it an invaluable tool in both academic and real-world scenarios.
