Solving Systems Of Equations Y = (1/2)x - 6 And X = -4

In mathematics, a system of equations is a set of two or more equations with the same variables. The solution to a system of equations is the set of values for the variables that make all of the equations true. In simpler terms, it's finding the point where the lines represented by the equations intersect on a graph. Solving systems of equations is a fundamental skill in algebra and has applications in various fields, including science, engineering, and economics. This article provides a comprehensive guide on how to solve systems of equations, focusing on a specific example to illustrate the process. The system of equations we'll be tackling is:

$$\begin{array}{l} y = \frac{1}{2}x - 6 \\ x = -4 \end{array}$$

This article will guide you through the process of finding the solution to this system, providing step-by-step instructions and explanations to help you grasp the underlying concepts. We'll explore the substitution method, a powerful technique for solving systems of equations, and demonstrate how it applies to this particular problem. By the end of this guide, you'll have a solid understanding of how to solve similar systems of equations and be well-equipped to tackle more complex problems.

Understanding the Problem

Before diving into the solution, it's essential to understand what we're trying to achieve. We have two equations:

$$\begin{aligned} y &= \frac{1}{2}x - 6 \\ x &= -4 \end{aligned}$$

The first equation, y = (1/2)x - 6, represents a linear relationship between x and y. It's in slope-intercept form, where 1/2 is the slope and -6 is the y-intercept. This means that for every 2 units we move to the right on the x-axis, the line goes up 1 unit on the y-axis, and the line crosses the y-axis at the point (0, -6). The second equation, x = -4, represents a vertical line that passes through the point (-4, 0) on the coordinate plane. All points on this line have an x-coordinate of -4, while the y-coordinate can be any real number.

The solution to this system of equations is the point (x, y) that satisfies both equations simultaneously. Graphically, this is the point where the two lines intersect. To find this point, we can use the substitution method. This method involves substituting the value of one variable from one equation into the other equation. In this case, we already have the value of x from the second equation, which makes the substitution method particularly straightforward. By substituting x = -4 into the first equation, we can solve for y. This will give us the y-coordinate of the solution. Once we have both the x and y coordinates, we can express the solution as an ordered pair (x, y), which represents the point of intersection of the two lines.

Solving the System of Equations Using Substitution

The substitution method is a powerful technique for solving systems of equations, and it's particularly useful when one of the equations is already solved for one of the variables. In our case, the second equation, x = -4, is already solved for x. This makes the substitution method the most efficient approach for finding the solution. The first step in the substitution method is to identify the equation that is already solved for one variable. In our case, it's x = -4. This equation tells us that the x-coordinate of the solution must be -4. Now, we can substitute this value of x into the first equation to find the corresponding value of y. This is the core of the substitution method: we replace one variable in one equation with its equivalent expression from the other equation.

Substituting x = -4 into the first equation, y = (1/2)x - 6, we get:

$$y = \frac{1}{2}(-4) - 6$$

Now, we simplify the equation to solve for y. First, we multiply 1/2 by -4:

$$y = -2 - 6$$

Then, we subtract 6 from -2:

$$y = -8$$

So, we have found that when x = -4, y = -8. This means that the solution to the system of equations is the ordered pair (-4, -8). This ordered pair represents the point where the two lines intersect on the coordinate plane. To verify our solution, we can substitute these values of x and y back into both original equations to ensure they hold true. This step is crucial for avoiding errors and ensuring the accuracy of our solution.

Verifying the Solution

After finding a solution to a system of equations, it's crucial to verify that the solution is correct. This step helps prevent errors and ensures that the values we found for the variables satisfy all the equations in the system. To verify the solution, we substitute the values of x and y back into the original equations. If the values make both equations true, then we have found the correct solution. In our case, the solution we found is x = -4 and y = -8. We will substitute these values into both equations to verify.

The first equation is y = (1/2)x - 6. Substituting x = -4 and y = -8, we get:

$$-8 = \frac{1}{2}(-4) - 6$$

Simplifying the right side of the equation:

$$-8 = -2 - 6$$

$$-8 = -8$$

This equation is true, so the solution satisfies the first equation. Now, we need to check the second equation, which is x = -4. Since we already know that x = -4 in our solution, this equation is automatically satisfied. Alternatively, we can substitute x = -4 into the equation:

$$-4 = -4$$

This equation is also true. Since the solution x = -4 and y = -8 satisfies both equations, we can confidently conclude that it is the correct solution to the system of equations. This verification step provides assurance that our solution is accurate and that we have correctly applied the substitution method.

The Solution and Its Implications

We have successfully solved the system of equations:

$$\begin{array}{l} y = \frac{1}{2}x - 6 \\ x = -4 \end{array}$$

Using the substitution method, we found that the solution is x = -4 and y = -8. This means that the point of intersection of the two lines represented by these equations is the ordered pair (-4, -8). The ordered pair (-4, -8) represents a specific location on the coordinate plane. The x-coordinate, -4, indicates the horizontal position, and the y-coordinate, -8, indicates the vertical position. This point is the only point that lies on both lines simultaneously, making it the unique solution to the system of equations.

In practical terms, this solution tells us the values of x and y that satisfy both relationships described by the equations. For example, if these equations represented real-world scenarios, such as the cost of a service based on time and a fixed fee, the solution would tell us the specific time and cost that meet both conditions. The implications of this solution depend on the context of the problem. In some cases, the solution might represent a break-even point, an equilibrium, or an optimal value. Understanding the solution in the context of the problem is crucial for making informed decisions and drawing meaningful conclusions.

Conclusion

In this comprehensive guide, we have explored the process of solving systems of equations using the substitution method. We started by understanding the problem, which involved two equations: y = (1/2)x - 6 and x = -4. We recognized that the second equation was already solved for x, making the substitution method an ideal approach.

We then proceeded to solve the system by substituting the value of x from the second equation into the first equation. This allowed us to solve for y, resulting in the solution y = -8. We expressed the solution as an ordered pair (-4, -8), which represents the point of intersection of the two lines on the coordinate plane.

To ensure the accuracy of our solution, we verified it by substituting the values of x and y back into both original equations. This step confirmed that our solution satisfied both equations, giving us confidence in our answer.

Finally, we discussed the implications of the solution, emphasizing that the ordered pair (-4, -8) represents the unique point that lies on both lines simultaneously. We also highlighted the importance of understanding the solution in the context of the problem, as it can provide valuable insights and inform decision-making.

Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications. By mastering techniques like the substitution method, you can confidently tackle a variety of problems and gain a deeper understanding of mathematical relationships. This guide has provided a solid foundation for solving systems of equations, and we encourage you to continue practicing and exploring more advanced techniques to further enhance your mathematical abilities.