Solving 3x + 4y = 24 And -x + 2y = 2: A Step-by-Step Guide
In the realm of mathematics, solving systems of equations is a fundamental skill, particularly within algebra. This article delves into the intricacies of solving a specific system:
3x + 4y = 24
-x + 2y = 2
We will explore various methods, providing a comprehensive understanding for students, educators, and anyone keen on enhancing their algebraic proficiency. Mastering systems of equations opens doors to more advanced mathematical concepts and real-world applications, making it an invaluable tool in various fields.
Understanding Systems of Equations
A system of equations is a collection of two or more equations with the same set of variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, it's the point where the lines represented by the equations intersect on a graph. The system presented here, with two linear equations and two variables (x and y), is a common type encountered in algebra.
Why are Systems of Equations Important?
Systems of equations are not just abstract mathematical concepts; they have practical applications in numerous fields, including:
- Engineering: Designing structures and circuits often involves solving systems of equations to ensure stability and efficiency.
- Economics: Modeling supply and demand, analyzing market equilibrium, and forecasting economic trends frequently require the use of systems of equations.
- Computer Science: Algorithms for optimization, such as linear programming, rely on solving systems of equations.
- Physics: Analyzing motion, forces, and energy in complex systems often involves solving multiple equations simultaneously.
Methods for Solving Systems of Equations
There are several methods to solve systems of equations, each with its strengths and weaknesses. For the system at hand, we will explore three primary methods:
- Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation.
- Elimination Method (also known as the Addition Method): This method involves manipulating the equations to eliminate one variable by adding or subtracting the equations.
- Graphical Method: This method involves graphing the equations and finding the point of intersection, which represents the solution.
1. The Substitution Method: A Step-by-Step Approach
The substitution method is a powerful technique for solving systems of equations. It's particularly useful when one equation can be easily solved for one variable in terms of the other. Let's apply this method to our system:
3x + 4y = 24 (Equation 1)
-x + 2y = 2 (Equation 2)
Step 1: Solve one equation for one variable
Looking at the equations, it seems easier to solve Equation 2 for x because the coefficient of x is -1. This minimizes the risk of dealing with fractions in the initial steps.
-x + 2y = 2
-x = 2 - 2y
x = 2y - 2 (Solving for x)
Step 2: Substitute the expression into the other equation
Now that we have x in terms of y, we substitute this expression (2y - 2) into Equation 1:
3x + 4y = 24
3(2y - 2) + 4y = 24 (Substituting x)
Step 3: Solve for the remaining variable
Next, we simplify and solve the resulting equation for y:
6y - 6 + 4y = 24
10y - 6 = 24
10y = 30
y = 3 (Solving for y)
Step 4: Substitute the value back to find the other variable
Now that we have the value of y, we substitute it back into either Equation 1, Equation 2, or the expression we derived for x (x = 2y - 2). Using the expression for x is often the most straightforward approach:
x = 2y - 2
x = 2(3) - 2 (Substituting y = 3)
x = 6 - 2
x = 4 (Solving for x)
Step 5: Verify the solution
It's crucial to verify the solution by substituting the values of x and y into both original equations:
For Equation 1:
3x + 4y = 24
3(4) + 4(3) = 24
12 + 12 = 24
24 = 24 (Verified)
For Equation 2:
-x + 2y = 2
-4 + 2(3) = 2
-4 + 6 = 2
2 = 2 (Verified)
Since the solution (x = 4, y = 3) satisfies both equations, it is the correct solution to the system.
2. The Elimination Method: A Systematic Approach
The elimination method, also known as the addition method, is another effective technique for solving systems of equations. This method involves manipulating the equations so that the coefficients of one variable are opposites, allowing you to eliminate that variable by adding the equations. Let's apply this method to our system:
3x + 4y = 24 (Equation 1)
-x + 2y = 2 (Equation 2)
Step 1: Manipulate the equations to match coefficients
Our goal is to make the coefficients of either x or y opposites. Looking at the equations, it seems easier to eliminate x. To do this, we can multiply Equation 2 by 3:
3 * (-x + 2y) = 3 * 2
-3x + 6y = 6 (Modified Equation 2)
Now we have:
3x + 4y = 24 (Equation 1)
-3x + 6y = 6 (Modified Equation 2)
Step 2: Add the equations to eliminate a variable
Now we add Equation 1 and the modified Equation 2:
(3x + 4y) + (-3x + 6y) = 24 + 6
10y = 30
Step 3: Solve for the remaining variable
Solve the resulting equation for y:
10y = 30
y = 3 (Solving for y)
Step 4: Substitute the value back to find the other variable
Substitute the value of y back into either Equation 1 or Equation 2 to solve for x. Let's use Equation 2:
-x + 2y = 2
-x + 2(3) = 2 (Substituting y = 3)
-x + 6 = 2
-x = -4
x = 4 (Solving for x)
Step 5: Verify the solution
As with the substitution method, it's essential to verify the solution by substituting the values of x and y into both original equations. We've already done this verification in the substitution method section, and we know that (x = 4, y = 3) satisfies both equations.
3. The Graphical Method: Visualizing the Solution
The graphical method provides a visual representation of the system of equations and its solution. This method involves graphing each equation on the same coordinate plane and finding the point where the lines intersect. The coordinates of the intersection point represent the solution to the system. Let's apply this method to our system:
3x + 4y = 24 (Equation 1)
-x + 2y = 2 (Equation 2)
Step 1: Convert equations to slope-intercept form (y = mx + b)
To graph the equations easily, we convert them to slope-intercept form, where m is the slope and b is the y-intercept.
For Equation 1:
3x + 4y = 24
4y = -3x + 24
y = (-3/4)x + 6
For Equation 2:
-x + 2y = 2
2y = x + 2
y = (1/2)x + 1
Step 2: Graph the equations
Now we graph both equations on the same coordinate plane. The slope-intercept form makes this straightforward. For Equation 1, the y-intercept is 6, and the slope is -3/4. For Equation 2, the y-intercept is 1, and the slope is 1/2.
When you graph these lines, you'll see they intersect at the point (4, 3).
Step 3: Identify the point of intersection
The point of intersection represents the solution to the system of equations. In this case, the lines intersect at (4, 3), meaning x = 4 and y = 3.
Step 4: Verify the solution
Again, it's essential to verify the solution by substituting the values of x and y into both original equations. As we've already done this, we know that (x = 4, y = 3) satisfies both equations.
Choosing the Right Method
Each method for solving systems of equations has its advantages and disadvantages:
- Substitution Method: Best when one equation can easily be solved for one variable.
- Elimination Method: Best when the coefficients of one variable are easily made opposites.
- Graphical Method: Best for visualizing the solution and understanding the relationship between the equations, but can be less accurate for non-integer solutions.
For the system 3x + 4y = 24 and -x + 2y = 2, the substitution and elimination methods are both efficient. The graphical method provides a visual confirmation of the solution.
Conclusion: Mastering Systems of Equations
Solving systems of equations is a crucial skill in mathematics with applications across various fields. By understanding and practicing the substitution, elimination, and graphical methods, you can confidently tackle these problems. Remember to always verify your solution to ensure accuracy. The solution to the system 3x + 4y = 24 and -x + 2y = 2 is indeed x = 4 and y = 3. Mastering this skill will undoubtedly enhance your mathematical prowess and problem-solving abilities.
This comprehensive guide has provided a thorough understanding of how to solve the system of equations 3x + 4y = 24 and -x + 2y = 2. Whether you prefer the algebraic precision of the substitution and elimination methods or the visual clarity of the graphical method, you now have the tools to confidently solve similar problems. Keep practicing, and you'll become proficient in this essential mathematical skill.
