In mathematics, a system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all equations in the system true simultaneously. Solving systems of equations is a fundamental concept in algebra and has applications in various fields, including science, engineering, economics, and computer science. Understanding how to solve these systems is crucial for tackling complex problems and modeling real-world scenarios.
This article delves into the process of solving a specific system of equations, providing a step-by-step guide and explaining the underlying concepts. We will focus on the given system:
y = (2/3)x + 3
x = -2
This system consists of two equations: a linear equation in slope-intercept form and a simple equation defining the value of x. Our goal is to find the value of y that corresponds to the given value of x and satisfies both equations. This involves substitution, a common technique used to solve systems of equations.
Before we dive into the solution, let's clarify what a system of equations is and why solving it matters. A system of equations is essentially a collection of equations that share the same variables. The solution to the system is the set of values that, when plugged into each equation, make each equation a true statement. Graphically, the solution represents the point(s) where the lines or curves represented by the equations intersect.
The importance of solving systems of equations lies in their ability to model real-world situations involving multiple constraints or relationships. For example, in economics, a system of equations might represent the supply and demand curves for a product, with the solution indicating the equilibrium price and quantity. In physics, a system of equations could describe the motion of multiple objects, with the solution providing their positions and velocities at a given time. Mastering the techniques to solve these systems is therefore essential for anyone working with mathematical models.
Now, let's tackle the specific system of equations presented. The system is:
y = (2/3)x + 3
x = -2
This system is particularly straightforward to solve because the second equation directly gives us the value of x. We can use this information to find the corresponding value of y using a method called substitution.
Step 1: Identify the Known Variable
The first step is to recognize that the second equation, x = -2, provides a specific value for the variable x. This is a crucial piece of information that simplifies the problem significantly. With the value of x known, we can proceed to the next step.
Step 2: Substitute the Known Value
Substitution is the process of replacing a variable in one equation with its known value from another equation. In this case, we substitute x = -2 into the first equation, y = (2/3)x + 3. This gives us:
y = (2/3)(-2) + 3
By substituting the value of x, we have transformed the first equation into an equation with only one variable, y. This makes it easy to solve for y.
Step 3: Simplify the Equation
Next, we simplify the equation to isolate y. First, multiply (2/3) by -2:
y = -4/3 + 3
Now, we need to add -4/3 and 3. To do this, we convert 3 to a fraction with a denominator of 3:
y = -4/3 + 9/3
Adding the fractions, we get:
y = 5/3
Step 4: State the Solution
The solution to the system of equations is the pair of values for x and y that satisfy both equations. We found that x = -2 and y = 5/3. Therefore, the solution to the system is:
(x, y) = (-2, 5/3)
This means that the point (-2, 5/3) is the intersection of the two lines represented by the equations in the system. We have successfully solved the system of equations using the substitution method.
To further understand the solution, let's visualize the system of equations graphically. The equation y = (2/3)x + 3 represents a line with a slope of 2/3 and a y-intercept of 3. The equation x = -2 represents a vertical line passing through the point (-2, 0) on the x-axis.
When we graph these two lines on the same coordinate plane, we see that they intersect at the point (-2, 5/3), which confirms our solution. The point of intersection is the graphical representation of the solution to the system of equations. It is the only point that lies on both lines, meaning it is the only pair of (x, y) values that satisfy both equations simultaneously.
Visualizing the solution graphically can provide a deeper understanding of the system of equations and the concept of a solution. It helps to see how the equations relate to each other and how their intersection represents the common solution.
While we used substitution to solve the given system, it's important to know that there are several methods for solving systems of equations, each with its strengths and weaknesses. The choice of method often depends on the specific system of equations and the ease with which one can apply the method.
Here are some common methods for solving systems of equations:
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Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable and allows you to solve for the remaining variable. We used this method to solve the given system of equations.
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Elimination Method (or Addition Method): This method involves manipulating the equations so that when they are added together, one of the variables is eliminated. This is done by multiplying one or both equations by constants so that the coefficients of one variable are opposites. Then, the equations are added, and the resulting equation is solved for the remaining variable.
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Graphing Method: This method involves graphing the equations on the same coordinate plane. The solution to the system is the point(s) where the graphs intersect. This method is useful for visualizing the solution, but it may not be accurate for systems with non-integer solutions.
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Matrix Methods: For larger systems of equations, matrix methods such as Gaussian elimination or using the inverse of a matrix can be more efficient. These methods are often used in computer programs to solve systems of equations.
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Numerical Methods: When dealing with nonlinear systems or systems that are difficult to solve analytically, numerical methods such as Newton's method or iterative methods can be used to approximate the solution.
Understanding these different methods allows you to choose the most appropriate technique for solving a particular system of equations efficiently.
Systems of equations are not just abstract mathematical concepts; they have numerous real-world applications across various disciplines. Being able to solve these systems is crucial for modeling and understanding many phenomena. Here are some examples of how systems of equations are used in different fields:
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Science and Engineering: In physics, systems of equations can describe the motion of objects, the flow of fluids, and the behavior of electrical circuits. In chemistry, they can be used to balance chemical equations and determine reaction rates. In engineering, they are used in structural analysis, circuit design, and control systems.
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Economics: Systems of equations are used extensively in economics to model supply and demand, market equilibrium, and macroeconomic models. They can help economists analyze the effects of different policies and predict economic trends.
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Computer Science: Systems of equations are used in computer graphics, game development, and optimization problems. They are also used in machine learning and data analysis to find patterns and relationships in data.
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Business and Finance: Systems of equations can be used to solve problems related to budgeting, investment analysis, and resource allocation. They can help businesses make informed decisions about pricing, production, and marketing.
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Everyday Life: Even in everyday situations, systems of equations can be helpful. For example, they can be used to determine the best deal when buying products with different pricing structures or to plan a budget with multiple income sources and expenses.
These are just a few examples of the many applications of systems of equations. The ability to solve these systems is a valuable skill that can be applied in various fields and situations.
In summary, we have explored the process of solving a system of equations, focusing on the system:
y = (2/3)x + 3
x = -2
We used the substitution method to find the solution, which is the pair of values (x, y) = (-2, 5/3). This means that when x is -2, y is 5/3, and both equations in the system are satisfied. We also discussed the graphical representation of the solution, which is the point of intersection of the lines represented by the equations.
Furthermore, we discussed other methods for solving systems of equations, including the elimination method, graphing method, matrix methods, and numerical methods. Each method has its advantages and is suitable for different types of systems. Understanding these methods provides a versatile toolkit for solving a wide range of problems.
Finally, we highlighted the numerous applications of systems of equations in various fields, including science, engineering, economics, computer science, business, and everyday life. This underscores the importance of mastering the techniques for solving these systems.
Solving systems of equations is a fundamental skill in mathematics with far-reaching applications. By understanding the concepts and methods discussed in this article, you can confidently tackle a wide range of problems and gain a deeper appreciation for the power of mathematical modeling.
Whether you are a student learning algebra, a professional working in a technical field, or simply someone interested in problem-solving, the ability to solve systems of equations is a valuable asset. Practice and familiarity with different methods will enhance your skills and allow you to apply these techniques effectively in various contexts.