Solving Systems Of Linear Equations Step-by-Step Guide
In this article, we will delve into the process of determining which ordered pair is a solution to a given system of linear equations. Understanding how to solve systems of equations is a fundamental concept in algebra and is essential for various applications in mathematics and other fields. We will explore a step-by-step approach to identify the correct solution by substituting the given ordered pairs into the equations and verifying if they satisfy both equations simultaneously.
Understanding Systems of Linear Equations
A system of linear equations consists of two or more linear equations involving the same variables. The solution to a system of linear equations is the set of values for the variables that make all the equations in the system true. Graphically, the solution represents the point(s) where the lines represented by the equations intersect. There are several methods to solve systems of linear equations, including substitution, elimination, and graphing. However, in this case, we are provided with potential solutions in the form of ordered pairs and can directly test them.
The Given System of Equations
We are given the following system of linear equations:
2x + 3y = 6
-3x + 5y = 10
Our goal is to determine which of the provided ordered pairs (A, B, C, and D) satisfies both equations. An ordered pair is written in the form (x, y), where x represents the value of the variable x, and y represents the value of the variable y.
The Proposed Solutions
We are given four potential solutions:
A. (0, 2) B. (2, 0) C. (3, 2) D. (2, 3)
To find the correct solution, we will substitute the x and y values from each ordered pair into both equations and check if the equations hold true.
Step-by-Step Solution
Step 1: Test Option A (0, 2)
Substitute x = 0 and y = 2 into the first equation:
2(0) + 3(2) = 6
0 + 6 = 6
6 = 6 (True)
The first equation is satisfied. Now, substitute x = 0 and y = 2 into the second equation:
-3(0) + 5(2) = 10
0 + 10 = 10
10 = 10 (True)
The second equation is also satisfied. Since the ordered pair (0, 2) satisfies both equations, it is a solution to the system.
Step 2: Test Option B (2, 0)
Substitute x = 2 and y = 0 into the first equation:
2(2) + 3(0) = 6
4 + 0 = 6
4 = 6 (False)
The first equation is not satisfied. Therefore, (2, 0) is not a solution to the system.
Step 3: Test Option C (3, 2)
Substitute x = 3 and y = 2 into the first equation:
2(3) + 3(2) = 6
6 + 6 = 6
12 = 6 (False)
The first equation is not satisfied. Therefore, (3, 2) is not a solution to the system.
Step 4: Test Option D (2, 3)
Substitute x = 2 and y = 3 into the first equation:
2(2) + 3(3) = 6
4 + 9 = 6
13 = 6 (False)
The first equation is not satisfied. Therefore, (2, 3) is not a solution to the system.
Conclusion
After testing each ordered pair, we found that only option A, (0, 2), satisfies both equations in the system. Therefore, the solution to the system of linear equations is (0, 2).
In summary, the ordered pair (0, 2) is a solution to the system of linear equations:
2x + 3y = 6
-3x + 5y = 10
This step-by-step approach of substituting the given ordered pairs into the equations and verifying if they hold true is a reliable method for solving this type of problem. Understanding and applying this method will help you solve similar problems involving systems of linear equations.
Additional Tips for Solving Systems of Linear Equations
Solving systems of linear equations is a fundamental skill in algebra, and there are several techniques you can use to find the solutions. In this article, we focused on testing given ordered pairs, but let's briefly explore other methods to enhance your problem-solving toolkit.
1. Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is already solved for one variable or can be easily manipulated. For example, consider the system:
y = 2x + 1
3x + y = 10
Here, the first equation is already solved for y. We can substitute the expression 2x + 1 for y in the second equation:
3x + (2x + 1) = 10
Simplifying and solving for x:
5x + 1 = 10
5x = 9
x = 9/5
Now, substitute the value of x back into the first equation to find y:
y = 2(9/5) + 1
y = 18/5 + 1
y = 23/5
Thus, the solution is (9/5, 23/5).
2. Elimination Method
The elimination method (also known as the addition method) involves adding or subtracting the equations in the system to eliminate one of the variables. This is often achieved by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. Consider the system:
2x + 3y = 7
4x - 3y = 5
Notice that the coefficients of y are already opposites (3 and -3). Adding the two equations eliminates y:
(2x + 3y) + (4x - 3y) = 7 + 5
6x = 12
x = 2
Substitute x = 2 into one of the original equations to find y:
2(2) + 3y = 7
4 + 3y = 7
3y = 3
y = 1
Thus, the solution is (2, 1).
3. Graphing Method
The graphing method involves plotting the lines represented by the equations on a coordinate plane. The solution to the system is the point(s) where the lines intersect. This method is particularly useful for visualizing the solutions and understanding the nature of the system (e.g., whether it has one solution, no solutions, or infinitely many solutions).
For example, consider the system:
y = x + 1
y = -x + 3
Plotting these lines on a graph, you'll find that they intersect at the point (1, 2). Thus, the solution is (1, 2).
4. Matrix Method
The matrix method is a more advanced technique that uses matrices to solve systems of linear equations. This method is particularly efficient for systems with many variables and equations. It involves representing the system as a matrix equation and using techniques such as Gaussian elimination or matrix inversion to find the solution.
For example, consider the system:
2x + y = 5
x - y = 1
This can be represented as the matrix equation:
| 2 1 | | x | = | 5 |
| 1 -1 | | y | | 1 |
Using matrix operations, you can solve for the variables x and y.
Tips for Choosing the Right Method
- If you are given potential solutions, as in the initial problem, testing each ordered pair is often the quickest method.
- If one of the equations is already solved for a variable or can be easily manipulated, the substitution method may be the most efficient.
- If the coefficients of one variable are opposites or can be easily made opposites, the elimination method is a good choice.
- The graphing method is useful for visualizing the solutions and understanding the nature of the system.
- The matrix method is best suited for larger systems with many variables and equations.
By mastering these techniques and understanding when to apply them, you'll be well-equipped to solve a wide range of systems of linear equations.
Real-World Applications of Systems of Linear Equations
Systems of linear equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can help you appreciate the practical significance of this topic and enhance your problem-solving skills.
1. Economics and Finance
In economics and finance, systems of linear equations are used to model and analyze supply and demand, market equilibrium, and investment strategies. For example, consider a simple supply and demand model:
Supply: Qs = aP + b
Demand: Qd = cP + d
Where:
- Qs is the quantity supplied
- Qd is the quantity demanded
- P is the price
- a, b, c, and d are constants
At equilibrium, the quantity supplied equals the quantity demanded (Qs = Qd). Solving this system of equations gives the equilibrium price and quantity. Similarly, in finance, systems of equations can be used to model portfolio optimization, where the goal is to maximize returns while minimizing risk.
2. Engineering and Physics
Engineering and physics heavily rely on systems of linear equations to solve problems related to circuits, mechanics, and fluid dynamics. For example, in electrical engineering, Kirchhoff's laws are used to analyze electrical circuits. These laws result in systems of linear equations that can be solved to find the currents and voltages in different parts of the circuit. Consider a simple circuit with two loops:
Loop 1: R1I1 + R2(I1 - I2) = V1
Loop 2: R2(I2 - I1) + R3I2 = V2
Where:
- I1 and I2 are the currents in the loops
- R1, R2, and R3 are the resistances
- V1 and V2 are the voltage sources
Solving this system of equations gives the currents I1 and I2. In mechanics, systems of equations are used to analyze forces and motion, while in fluid dynamics, they can model fluid flow and pressure distributions.
3. Computer Graphics and Game Development
Computer graphics and game development use systems of linear equations for transformations, projections, and rendering. For example, to rotate or scale an object in 3D space, matrices are used, and these matrix operations often involve solving systems of linear equations. Consider a simple 2D rotation:
x' = x cos(θ) - y sin(θ)
y' = x sin(θ) + y cos(θ)
Where:
- (x, y) are the original coordinates
- (x', y') are the rotated coordinates
- θ is the rotation angle
This system of equations transforms the coordinates of a point. In game development, systems of equations are also used for collision detection, pathfinding, and artificial intelligence.
4. Chemistry
In chemistry, systems of linear equations are used to balance chemical equations and solve stoichiometry problems. Balancing chemical equations involves finding the coefficients that make the number of atoms of each element the same on both sides of the equation. For example, consider the equation:
aC3H8 + bO2 → cCO2 + dH2O
Balancing this equation involves solving a system of linear equations for the coefficients a, b, c, and d.
5. Resource Allocation and Optimization
Resource allocation and optimization problems often involve systems of linear equations. For example, a company might want to allocate its resources (e.g., labor, materials, capital) to maximize profit or minimize cost. Linear programming, a technique used in operations research, relies heavily on solving systems of linear equations and inequalities. Consider a simple production planning problem:
Maximize: P = aX + bY
Subject to:
cX + dY ≤ R1
eX + fY ≤ R2
X, Y ≥ 0
Where:
- P is the profit
- X and Y are the quantities of two products
- a and b are the profits per unit of X and Y
- R1 and R2 are the resource constraints
- c, d, e, and f are the resource requirements per unit of X and Y
Solving this system helps determine the optimal production quantities X and Y.
6. Data Analysis and Statistics
In data analysis and statistics, systems of linear equations are used for regression analysis, curve fitting, and solving statistical models. For example, linear regression involves finding the best-fit line through a set of data points. This often involves solving a system of linear equations to find the coefficients of the line. Consider a simple linear regression model:
y = mx + b
Where:
- y is the dependent variable
- x is the independent variable
- m is the slope
- b is the y-intercept
Solving for m and b involves using a system of equations derived from the data.
7. Cryptography
Cryptography, the science of secure communication, uses systems of linear equations in various encryption and decryption algorithms. While modern cryptography often employs more complex mathematical structures, basic encryption techniques can be based on linear transformations and systems of equations.
By understanding these diverse applications, you can see that systems of linear equations are a powerful tool for modeling and solving real-world problems. Whether it's balancing chemical equations, designing electrical circuits, or optimizing resource allocation, the ability to solve systems of linear equations is a valuable skill in many fields.
Conclusion
In conclusion, solving systems of linear equations is a fundamental skill with broad applications across various disciplines. Whether you are testing potential solutions, using substitution or elimination, or applying more advanced techniques like matrices, mastering these methods will enhance your problem-solving abilities. From economics and engineering to computer graphics and cryptography, the ability to model and solve systems of linear equations is invaluable. By understanding the theory and practice of these techniques, you'll be well-equipped to tackle complex problems and make informed decisions in a wide range of contexts.
Remember, the ordered pair (0, 2) is the solution to the given system of linear equations, and this step-by-step approach can be applied to similar problems. Keep practicing and exploring the real-world applications to deepen your understanding and skills in this essential mathematical area.
