Solving Systems Of Equations 4x + 5y = 7 And 3x - 2y = -12
This article provides a step-by-step guide on how to solve the system of equations:
4x + 5y = 7
3x - 2y = -12
We will explore the methodology behind solving such systems, emphasizing the elimination method and providing clear explanations to ensure a comprehensive understanding. This method involves manipulating the equations to eliminate one variable, allowing us to solve for the other. By applying this method systematically, we can determine the values of 'x' and 'y' that satisfy both equations simultaneously. Let's delve into the process and break down each step involved in finding the solution.
Understanding Systems of Equations
Before diving into the solution, it's crucial to understand what a system of equations represents. 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 the equations true simultaneously. In simpler terms, we are looking for the point (x, y) where the lines represented by the equations intersect on a graph. Systems of equations arise in various mathematical and real-world applications, from calculating mixtures in chemistry to determining optimal pricing strategies in economics. The ability to solve these systems efficiently is a fundamental skill in mathematics and its related fields.
There are several methods to solve systems of equations, including graphing, substitution, and elimination. The graphing method involves plotting the equations on a coordinate plane and finding the point of intersection. This method is visually intuitive but can be less accurate for non-integer solutions. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This method is useful when one equation is easily solved for one variable. The elimination method, which we will use in this article, involves manipulating the equations to eliminate one variable by adding or subtracting the equations. This method is particularly efficient when the coefficients of one variable are opposites or can be easily made opposites. Each method has its strengths and weaknesses, and the choice of method often depends on the specific system of equations being solved. In this article, we will focus on the elimination method, which is a powerful technique for solving linear systems of equations.
H2: The Elimination Method
The elimination method is a powerful technique for solving systems of equations. It involves manipulating the equations so that the coefficients of one variable are opposites. When the equations are added, this variable is eliminated, leaving an equation with only one variable. This resulting equation can then be easily solved, and the value of the solved variable can be substituted back into one of the original equations to find the value of the other variable. The elimination method is particularly useful when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant. The key is to strategically choose the multipliers to make the elimination process as straightforward as possible.
Step 1: Multiplying Equations to Achieve Opposite Coefficients
The first step in the elimination method is to multiply one or both equations by a constant so that the coefficients of either 'x' or 'y' are opposites. In our system:
4x + 5y = 7
3x - 2y = -12
We can choose to eliminate either 'x' or 'y'. Let's choose to eliminate 'x'. To do this, we need to find the least common multiple (LCM) of the coefficients of 'x', which are 4 and 3. The LCM of 4 and 3 is 12. We will multiply the first equation by 3 and the second equation by -4 so that the coefficients of 'x' become 12 and -12, respectively. This will allow us to eliminate 'x' when we add the equations together.
Multiplying the first equation by 3, we get:
3 * (4x + 5y) = 3 * 7
12x + 15y = 21
Multiplying the second equation by -4, we get:
-4 * (3x - 2y) = -4 * (-12)
-12x + 8y = 48
Now we have a new system of equations:
12x + 15y = 21
-12x + 8y = 48
Notice that the coefficients of 'x' are now opposites (12 and -12), which sets the stage for the next step in the elimination method. This strategic multiplication is crucial for simplifying the system and making it easier to solve. By carefully choosing the multipliers, we can ensure that one variable will be eliminated when the equations are added, leading us closer to the solution.
Step 2: Adding the Equations
Now that we have opposite coefficients for 'x', we can add the two equations together. Adding the equations term by term, we have:
(12x + 15y) + (-12x + 8y) = 21 + 48
Combining like terms, the 'x' terms cancel out (12x - 12x = 0), leaving us with:
15y + 8y = 69
23y = 69
This step is the core of the elimination method. By strategically manipulating the equations in the previous step, we have successfully eliminated one variable ('x'), resulting in a single equation with only one variable ('y'). This simplified equation can now be easily solved for 'y'. The cancellation of the 'x' terms is a direct result of our careful choice of multipliers in the previous step, which ensured that the coefficients of 'x' were opposites. The resulting equation, 23y = 69, is a linear equation in one variable, which is straightforward to solve using basic algebraic techniques.
Step 3: Solving for y
To solve for 'y', we divide both sides of the equation 23y = 69 by 23:
23y / 23 = 69 / 23
y = 3
So, we have found that y = 3. This is one part of the solution to the system of equations. We now know the y-coordinate of the point where the two lines intersect. The next step is to substitute this value of 'y' back into one of the original equations to solve for 'x'. This process of substitution allows us to determine the corresponding value of 'x' that satisfies both equations simultaneously. The value y = 3 is a critical piece of the puzzle, and substituting it back into one of the original equations will reveal the value of 'x', completing the solution to the system of equations.
Step 4: Substituting y to Solve for x
Now that we have the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first equation:
4x + 5y = 7
Substituting y = 3, we get:
4x + 5(3) = 7
4x + 15 = 7
Subtracting 15 from both sides:
4x = 7 - 15
4x = -8
Dividing both sides by 4:
x = -8 / 4
x = -2
So, we have found that x = -2. This gives us the x-coordinate of the solution. Combining this with the value of y we found earlier, we now have the complete solution to the system of equations. Substituting the value of y back into one of the original equations is a common technique in solving systems of equations. It allows us to leverage the value we found for one variable to determine the value of the other variable, ultimately leading us to the solution that satisfies both equations simultaneously. The careful substitution and algebraic manipulation in this step are essential for accurately determining the value of 'x'.
H3: The Solution
Therefore, the solution to the system of equations is:
x = -2, y = 3
This can be written as the ordered pair (-2, 3). This ordered pair represents the point of intersection of the two lines represented by the equations in the system. It is the unique point that satisfies both equations simultaneously. To verify the solution, we can substitute these values back into both original equations to ensure that they hold true. This verification step is crucial to confirm the accuracy of our solution and to catch any potential errors in our calculations. The solution (-2, 3) is the final answer to the problem, representing the values of 'x' and 'y' that make both equations in the system true.
Verification
Let's verify the solution by substituting x = -2 and y = 3 into both original equations:
Equation 1:
4x + 5y = 7
4(-2) + 5(3) = 7
-8 + 15 = 7
7 = 7 (True)
Equation 2:
3x - 2y = -12
3(-2) - 2(3) = -12
-6 - 6 = -12
-12 = -12 (True)
Since the solution satisfies both equations, we have confirmed that our answer is correct. Verification is an essential step in solving any mathematical problem, especially systems of equations. By substituting the solution back into the original equations, we can ensure that the values we found are indeed the correct ones. This process helps to eliminate errors and provides confidence in the accuracy of our solution. In this case, the verification step confirms that x = -2 and y = 3 is the correct solution to the system of equations.
H2: Conclusion
In this article, we have thoroughly explored the process of solving the system of equations:
4x + 5y = 7
3x - 2y = -12
using the elimination method. We systematically manipulated the equations to eliminate one variable, solved for the remaining variable, and then substituted back to find the value of the other variable. The solution we found is x = -2 and y = 3, or the ordered pair (-2, 3). We also verified this solution by substituting the values back into the original equations, confirming its accuracy. The ability to solve systems of equations is a fundamental skill in mathematics and has numerous applications in various fields. The elimination method is a powerful technique for solving linear systems, and understanding this method provides a solid foundation for tackling more complex mathematical problems. Mastering these techniques can greatly enhance problem-solving abilities and provides a valuable tool for mathematical analysis.
This step-by-step approach not only provides the solution but also enhances understanding of the underlying concepts. The elimination method is a cornerstone of algebra and a critical tool for solving a wide array of problems. From linear programming to circuit analysis, the applications of systems of equations are vast and varied. By understanding the mechanics of the elimination method, one gains the ability to approach these problems with confidence and accuracy. The systematic approach outlined in this article ensures that the solution is not only correct but also understood, promoting a deeper appreciation for the mathematical principles involved. This understanding is crucial for building a strong foundation in mathematics and for applying these concepts in real-world scenarios. The ability to solve systems of equations is a valuable asset in any quantitative discipline.
