Solving System Of Equations A Step By Step Guide
This article delves into the step-by-step process of solving a system of linear equations. Solving system of equations is a fundamental concept in mathematics with wide-ranging applications in various fields such as engineering, physics, economics, and computer science. We will explore a specific system of equations and demonstrate how to find its solution using algebraic techniques. Understanding how to solve system of equations is crucial for anyone pursuing studies or careers in STEM fields. Before we dive into the solution, let's first understand what a system of linear equations is and why it's important.
A system of linear equations is a set of two or more linear equations containing the same variables. A linear equation is an equation in which the highest power of the variable is 1. The solution to a system of linear equations is a set of values for the variables that satisfies all the equations simultaneously. In other words, it's the point where all the lines represented by the equations intersect. There are several methods to solve system of equations, including substitution, elimination, and matrix methods. The choice of method depends on the specific system of equations and the ease of application.
Systems of equations arise in numerous real-world scenarios. For instance, in economics, supply and demand curves can be represented as linear equations, and their intersection point determines the market equilibrium. In physics, systems of equations can be used to analyze the motion of objects under multiple forces. In computer graphics, systems of equations are used to perform transformations on objects in 3D space. Therefore, the ability to solve system of equations is an essential skill for professionals in many different disciplines. Let's begin by stating the system of equations we aim to solve.
The Given System of Equations
We are given the following system of equations:
-3x + 3y = -5/8
x - 4z = -5/3
-4y + z = 0
This system consists of three linear equations with three variables: x, y, and z. Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. To solve system of equations effectively, we can employ various strategies, including substitution, elimination, or matrix methods. In this case, we'll use a combination of substitution and elimination to systematically isolate the variables and find their values. The first step is to label the equations for easy reference.
Let's label the equations as follows:
Equation 1: -3x + 3y = -5/8 Equation 2: x - 4z = -5/3 Equation 3: -4y + z = 0
Now that we have labeled the equations, we can start to manipulate them to eliminate variables. One common strategy is to solve one equation for one variable and then substitute that expression into another equation. This reduces the number of variables in the second equation, making it easier to solve. We can also multiply equations by constants to make the coefficients of one variable the same, allowing us to eliminate that variable by adding or subtracting the equations. The key to solve system of equations is to choose the most efficient method to minimize the amount of calculation needed.
Step-by-Step Solution
Step 1: Solving for z in Equation 3
We can start by solving Equation 3 for z:
-4y + z = 0 z = 4y
This is a crucial step because it expresses z in terms of y. This will allow us to substitute this expression for z in other equations, effectively reducing the number of variables. This substitution method is a powerful technique to solve system of equations. By expressing one variable in terms of another, we can simplify the system and make it easier to handle. This approach is particularly useful when one of the equations has a variable with a coefficient of 1, as in this case. Substituting this expression into other equations will help us eliminate z and get closer to the solution.
Step 2: Substituting z in Equation 2
Now, substitute z = 4y into Equation 2:
x - 4(4y) = -5/3 x - 16y = -5/3
This substitution is a key step in solve system of equations. We have successfully eliminated z from Equation 2, leaving us with an equation involving only x and y. This equation, along with Equation 1, now forms a system of two equations with two variables, which is much easier to solve. The goal is to reduce the number of variables in each equation until we can isolate one variable and find its value. The ability to strategically substitute variables is a fundamental skill in solving systems of equations. This process of substitution is a cornerstone of algebraic manipulation.
Step 3: Expressing x in terms of y
Express x in terms of y:
x = 16y - 5/3
By isolating x, we now have an expression for x in terms of y. This will allow us to substitute this expression into Equation 1, further reducing the complexity of the system. The ability to manipulate equations and isolate variables is a crucial skill in solving systems of equations. By expressing x in terms of y, we are setting the stage for the next substitution, which will allow us to solve for y directly. The systematic manipulation of equations is the key to solve system of equations efficiently.
Step 4: Substituting x in Equation 1
Substitute x = 16y - 5/3 into Equation 1:
-3(16y - 5/3) + 3y = -5/8 -48y + 5 + 3y = -5/8 -45y = -5/8 - 5 -45y = -5/8 - 40/8 -45y = -45/8 y = (-45/8) / (-45) y = 1/8
This substitution is a critical step in solve system of equations. By substituting the expression for x in terms of y into Equation 1, we have successfully eliminated x and obtained an equation involving only y. This allows us to directly solve for y. Once we have the value of y, we can then substitute it back into other equations to find the values of x and z. This process of substitution and solving for variables one at a time is a fundamental strategy in solving systems of equations. Solving for y is a significant milestone in the process.
Step 5: Solving for x
Now that we have y = 1/8, we can substitute it back into the equation x = 16y - 5/3:
x = 16(1/8) - 5/3 x = 2 - 5/3 x = 6/3 - 5/3 x = 1/3
With the value of y determined, substituting it back into the expression for x allows us to find the value of x. This is a common technique in solve system of equations, where we use previously found values to determine the remaining variables. By working systematically, we can break down the problem into smaller, more manageable steps. Finding the value of x is another step closer to the complete solution of the system. This step demonstrates the interconnectedness of the equations and the importance of using previously found values to find new ones.
Step 6: Solving for z
Substitute y = 1/8 into z = 4y:
z = 4(1/8) z = 1/2
Finally, we substitute the value of y into the equation z = 4y to find the value of z. This completes the solution process. We have now found the values of all three variables: x, y, and z. This final substitution is a crucial step in solve system of equations, as it allows us to find the last unknown variable. The systematic approach of substitution and solving for variables one at a time has led us to the complete solution of the system. With the values of x, y, and z determined, we can now state the solution as an ordered triple.
The Solution
Therefore, the solution to the system of equations is:
(x, y, z) = (1/3, 1/8, 1/2)
This solution represents the point where all three planes represented by the equations intersect. To verify that this is the correct solution, we can substitute these values back into the original equations and check that they satisfy all three equations. This is a crucial step in solve system of equations, as it ensures that our solution is accurate. By substituting the values back into the original equations, we can confirm that the left-hand side equals the right-hand side for each equation. This verification process gives us confidence in the correctness of our solution.
Verification
Let's verify the solution by substituting the values of x, y, and z into the original equations:
Equation 1: -3x + 3y = -5/8 -3(1/3) + 3(1/8) = -1 + 3/8 = -8/8 + 3/8 = -5/8 (Correct)
Equation 2: x - 4z = -5/3 1/3 - 4(1/2) = 1/3 - 2 = 1/3 - 6/3 = -5/3 (Correct)
Equation 3: -4y + z = 0 -4(1/8) + 1/2 = -1/2 + 1/2 = 0 (Correct)
The solution (1/3, 1/8, 1/2) satisfies all three equations. This confirms that our solution is correct. The process of verification is an essential part of solve system of equations, as it helps to catch any errors that may have occurred during the solution process. By substituting the solution back into the original equations, we can ensure that the values we have found are indeed the correct ones. This verification step provides a final check and ensures the accuracy of our solution.
Conclusion
In conclusion, we have successfully solved the system of equations:
-3x + 3y = -5/8
x - 4z = -5/3
-4y + z = 0
using a combination of substitution and elimination methods. The solution we found is (x, y, z) = (1/3, 1/8, 1/2). This article has demonstrated a systematic approach to solve system of equations. By labeling equations, substituting variables, and simplifying expressions, we can effectively find the solution to complex systems of equations. The ability to solve systems of equations is a valuable skill in many fields, and this step-by-step guide provides a solid foundation for further exploration of this topic. Remember, the key to success is to practice and apply these techniques to various problems.
