Solving Systems Of Equations Y=-3x-2 And 5x+2y=15 A Comprehensive Guide
Introduction to Systems of Equations
In the realm of 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. These systems are fundamental in various fields, including engineering, physics, economics, and computer science, as they help model and solve real-world problems involving multiple constraints and variables.
Understanding how to solve systems of equations is a critical skill in algebra and beyond. There are several methods available for solving these systems, each with its own advantages and applications. The most common methods include substitution, elimination (also known as the addition method), and graphical methods. This article will delve into these methods, providing step-by-step explanations and examples to help you master the art of solving systems of equations.
Before diving into the solution methods, it's essential to grasp the concept of a solution to a system of equations. A solution is an ordered pair (in the case of two variables) or an ordered triple (in the case of three variables), etc., that satisfies all equations in the system. Graphically, the solution to a system of two equations represents the point(s) where the lines or curves intersect. This intersection point is the visual representation of the values that make both equations true. Systems of equations can have one solution, no solution, or infinitely many solutions, depending on the relationship between the equations. Recognizing the type of solution is a crucial aspect of solving systems of equations, as it provides insights into the nature of the problem and its potential applications.
Methods for Solving Systems of Equations
1. Substitution Method
The substitution method is a powerful algebraic technique for solving systems of equations. This method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable. The substitution method is particularly useful when one of the equations is already solved for one variable or when it is easy to isolate one variable in one of the equations. For instance, if you have an equation like y = 2x + 3, substitution is a natural choice. This method is widely applicable and forms a cornerstone of algebraic problem-solving.
The steps involved in the substitution method are straightforward but require careful attention to detail. First, choose one equation and solve it for one variable. It's often best to select the equation and variable that will result in the simplest expression. Next, substitute the expression you found in the first step into the other equation. This will yield a new equation with only one variable. Solve this new equation for the remaining variable. Finally, substitute the value you found back into either of the original equations to find the value of the other variable. Be sure to check your solution by substituting both values into both original equations to ensure they are satisfied. This step is crucial for verifying the accuracy of your solution and preventing errors.
To illustrate, consider the system:
y = 2x + 1
3x + y = 10
Here, the first equation is already solved for y. Substitute 2x + 1 for y in the second equation: 3x + (2x + 1) = 10. Simplify and solve for x: 5x + 1 = 10, 5x = 9, x = 9/5. Now, substitute x = 9/5 back into the first equation: y = 2(9/5) + 1 = 18/5 + 1 = 23/5. Thus, the solution is x = 9/5 and y = 23/5. This step-by-step approach ensures clarity and accuracy when using the substitution method, making it an invaluable tool in your mathematical arsenal.
2. Elimination Method
The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. This method involves manipulating the equations in the system so that when they are added together, one of the variables is eliminated. This is achieved by multiplying one or both equations by a constant so that the coefficients of one of the variables are opposites. Once the equations are in this form, adding them together eliminates one variable, resulting in a single equation with one variable that can be easily solved. The elimination method is particularly useful when the equations are in standard form (Ax + By = C) and when the coefficients of one of the variables are easily made opposites. This method is a cornerstone of linear algebra and is widely used in various mathematical and scientific applications.
The key to the elimination method lies in the strategic manipulation of the equations. The first step is to examine the system and identify which variable would be easiest to eliminate. This often involves looking for variables with coefficients that are multiples of each other or that have opposite signs. Next, multiply one or both equations by a constant so that the coefficients of the chosen variable become opposites. For example, if you have the system 2x + 3y = 7 and 4x - y = 1, you could multiply the second equation by 3 to make the coefficients of y opposites: 2x + 3y = 7 and 12x - 3y = 3. Once the coefficients are opposites, add the equations together. This will eliminate one variable, leaving you with a single equation in one variable. Solve this equation for the remaining variable, and then substitute the value back into one of the original equations to find the value of the other variable. Always remember to check your solution by substituting both values into both original equations to ensure they are satisfied.
For example, consider the system:
3x + 2y = 7
5x - 2y = 1
In this case, the coefficients of y are already opposites (2 and -2). Adding the equations together eliminates y: (3x + 2y) + (5x - 2y) = 7 + 1, which simplifies to 8x = 8, and thus x = 1. Substitute x = 1 back into the first equation: 3(1) + 2y = 7, which gives 2y = 4, and y = 2. Therefore, the solution is x = 1 and y = 2. This clear, methodical approach makes the elimination method a reliable and efficient way to solve systems of equations, highlighting its importance in mathematical problem-solving.
3. Graphical Method
The graphical method offers a visual approach to solving systems of equations, particularly useful for linear systems. This method involves graphing each equation in the system on the same coordinate plane. The solution to the system is the point or points where the graphs intersect. Each line on the graph represents the set of all solutions for its respective equation, and the intersection point represents the solution that satisfies both equations simultaneously. The graphical method provides an intuitive understanding of the solution and is especially helpful for systems with two variables, where the equations represent lines. This method is widely used for visualizing solutions and can provide a quick estimate of the solution even when an exact algebraic solution is not immediately apparent.
To effectively use the graphical method, each equation must first be graphed accurately. This typically involves rewriting the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Once in this form, plotting the lines becomes straightforward. Start by plotting the y-intercept and then use the slope to find additional points on the line. Connect these points to draw the line. Repeat this process for each equation in the system. After graphing all equations, identify the point(s) of intersection. The coordinates of the intersection point(s) represent the solution to the system. If the lines intersect at a single point, there is one unique solution. If the lines are parallel and do not intersect, the system has no solution. If the lines are coincident (the same line), the system has infinitely many solutions. This visual interpretation of the solution is a key advantage of the graphical method.
For example, consider the system:
y = x + 1
y = -x + 3
Graphing these two lines on the same coordinate plane shows that they intersect at the point (1, 2). Therefore, the solution to the system is x = 1 and y = 2. The graphical method not only provides the solution but also gives a clear visual representation of the relationship between the equations. It is an invaluable tool for understanding systems of equations and can be particularly useful for students learning these concepts. While the graphical method is often used for linear systems, it can also be applied to non-linear systems, although the graphs may be more complex and require careful plotting. Overall, the graphical method is a fundamental technique in mathematics that enhances both understanding and problem-solving skills.
Solving the Given System of Equations
Now, let's apply these methods to solve the given system of equations:
y = -3x - 2
5x + 2y = 15
This system presents a classic example where the substitution method can be effectively employed due to the first equation already being solved for y. However, we will also demonstrate the elimination method to provide a comprehensive understanding of different approaches to solving the same problem.
1. Solving by Substitution
Since the first equation is y = -3x - 2, we can substitute this expression for y into the second equation:
5x + 2(-3x - 2) = 15
Now, distribute the 2:
5x - 6x - 4 = 15
Combine like terms:
-x - 4 = 15
Add 4 to both sides:
-x = 19
Multiply both sides by -1 to solve for x:
x = -19
Now that we have the value of x, we can substitute it back into the first equation to find the value of y:
y = -3(-19) - 2
y = 57 - 2
y = 55
Thus, the solution to the system using the substitution method is x = -19 and y = 55. This step-by-step approach highlights the efficiency of the substitution method when one equation is already solved for a variable.
2. Solving by Elimination
To solve the same system using the elimination method, we first need to align the variables in both equations. The given system is:
y = -3x - 2
5x + 2y = 15
Rewrite the first equation in the standard form (Ax + By = C):
3x + y = -2
5x + 2y = 15
Now, we can eliminate y by multiplying the first equation by -2:
-2(3x + y) = -2(-2)
-6x - 2y = 4
Our system now looks like this:
-6x - 2y = 4
5x + 2y = 15
Add the two equations together:
(-6x - 2y) + (5x + 2y) = 4 + 15
-x = 19
Multiply both sides by -1 to solve for x:
x = -19
Substitute x = -19 back into one of the original equations, let's use the first one:
3(-19) + y = -2
-57 + y = -2
Add 57 to both sides:
y = 55
Therefore, the solution to the system using the elimination method is also x = -19 and y = 55. This demonstrates the consistency of the solution, regardless of the method used. The elimination method, in this case, required an extra step of rearranging the equations but provided a clear path to the solution through the strategic elimination of a variable.
Verifying the Solution
To ensure the accuracy of our solution, it is crucial to verify the values of x and y in both original equations. Our solution is x = -19 and y = 55. The original system of equations is:
y = -3x - 2
5x + 2y = 15
Let's substitute x = -19 and y = 55 into the first equation:
55 = -3(-19) - 2
55 = 57 - 2
55 = 55
The first equation is satisfied.
Now, substitute x = -19 and y = 55 into the second equation:
5(-19) + 2(55) = 15
-95 + 110 = 15
15 = 15
The second equation is also satisfied.
Since both equations are satisfied by the values x = -19 and y = 55, we can confidently conclude that this is the correct solution to the system of equations. Verification is a vital step in the problem-solving process, as it ensures that the solution is accurate and that no algebraic errors were made during the solving process. This practice reinforces the understanding of the solution concept and builds confidence in mathematical abilities.
Conclusion
Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications across various fields. This article has explored the main methods for solving these systems, including the substitution method, the elimination method, and the graphical method. Each method offers a unique approach, and the choice of method often depends on the specific characteristics of the system. The substitution method is particularly effective when one equation is already solved for a variable, while the elimination method shines when equations are in standard form and coefficients can be easily manipulated to eliminate a variable. The graphical method provides a visual representation of the solution and is especially useful for linear systems.
We demonstrated these methods by solving the system:
y = -3x - 2
5x + 2y = 15
Using both the substitution and elimination methods, we found the solution to be x = -19 and y = 55. We also emphasized the importance of verifying the solution by substituting the values back into the original equations to ensure accuracy. This verification step is crucial for building confidence in your problem-solving abilities and preventing errors.
Mastering these techniques empowers you to tackle more complex mathematical problems and apply these skills in real-world scenarios. Whether you are solving problems in physics, engineering, economics, or computer science, a strong understanding of systems of equations is invaluable. Continue to practice these methods with various systems of equations to solidify your understanding and enhance your problem-solving skills. Remember, the key to success in mathematics is consistent practice and a solid grasp of fundamental concepts.
