Solving 2x+3y=13 And 4x-y=-2 Finding X And Y Values
In the realm of mathematics, solving systems of equations is a fundamental skill. These systems often arise in various real-world applications, from engineering and physics to economics and computer science. A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values for the variables that satisfy all the equations simultaneously. In this article, we will delve into the process of solving a system of two linear equations with two variables, specifically the system:
- 2x + 3y = 13
- 4x - y = -2
We will explore different methods to find the values of x and y that satisfy both equations. Mastering these techniques will provide a solid foundation for tackling more complex mathematical problems.
Understanding the Problem
Before we dive into the solution, let's first understand what we are trying to achieve. We have two equations, each representing a straight line on a graph. The solution to the system is the point where these two lines intersect. In other words, we are looking for the x and y coordinates that lie on both lines simultaneously. There are several methods to find this intersection point, including substitution, elimination, and graphical methods. In this article, we will primarily focus on the substitution and elimination methods, as they are the most commonly used and efficient for solving such systems.
The significance of solving systems of equations extends far beyond the classroom. In various fields, these systems model real-world scenarios where multiple variables are interconnected. For instance, in economics, they can represent supply and demand curves, and the solution indicates the equilibrium price and quantity. In physics, they can describe the motion of objects under multiple forces, and the solution reveals the resulting position and velocity. Understanding how to solve these systems is therefore a crucial skill for anyone pursuing a career in these areas.
Method 1: Substitution
Step 1: Isolate one variable in one equation.
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Let's start by isolating y in the second equation, 4x - y = -2. To do this, we can add y to both sides and add 2 to both sides, resulting in:
4x + 2 = y
Now we have y expressed in terms of x. This expression will be crucial for the next step.
Step 2: Substitute the expression into the other equation.
Now that we have y = 4x + 2, we can substitute this expression into the first equation, 2x + 3y = 13. This substitution will eliminate y from the first equation, leaving us with an equation in terms of x only:
2x + 3(4x + 2) = 13
This step is the heart of the substitution method. By replacing y with its equivalent expression in terms of x, we've reduced the problem to a single equation with one variable, which is much easier to solve.
Step 3: Solve for the remaining variable.
Let's solve the equation for x. First, distribute the 3:
2x + 12x + 6 = 13
Combine like terms:
14x + 6 = 13
Subtract 6 from both sides:
14x = 7
Divide both sides by 14:
x = 7/14 = 1/2
We have now found the value of x. This is a significant milestone in solving the system. We know one coordinate of the intersection point.
Step 4: Substitute the value back to find the other variable.
Now that we have x = 1/2, we can substitute this value back into either of the original equations or the expression we found for y in terms of x. Let's use the expression y = 4x + 2, as it's already isolated for y:
y = 4(1/2) + 2
y = 2 + 2
y = 4
We have found the value of y. Now we have both coordinates of the solution.
Step 5: Verify the solution.
To ensure our solution is correct, we should verify that the values x = 1/2 and y = 4 satisfy both original equations. Let's plug these values into the first equation:
2(1/2) + 3(4) = 1 + 12 = 13
The first equation is satisfied. Now let's check the second equation:
4(1/2) - 4 = 2 - 4 = -2
The second equation is also satisfied. Therefore, our solution is correct.
Result of Substitution Method
Thus, the solution to the system of equations using the substitution method is x = 1/2 and y = 4. This means the two lines represented by the equations intersect at the point (1/2, 4) on the coordinate plane.
Method 2: Elimination
Step 1: Multiply equations to make coefficients match.
The elimination method involves manipulating the equations so that the coefficients of one variable are opposites or the same. In this case, let's eliminate y. We can multiply the second equation (4x - y = -2) by 3 so that the coefficient of y becomes -3, which is the opposite of the coefficient of y in the first equation (2x + 3y = 13):
3(4x - y) = 3(-2)
12x - 3y = -6
Now we have two equations:
2x + 3y = 13
12x - 3y = -6
The coefficients of y are now opposites, setting the stage for the next step.
Step 2: Add or subtract the equations to eliminate a variable.
Now that the coefficients of y are opposites, we can add the two equations together. This will eliminate y from the resulting equation:
(2x + 3y) + (12x - 3y) = 13 + (-6)
14x = 7
By adding the equations, we've successfully eliminated y, leaving us with a single equation in terms of x.
Step 3: Solve for the remaining variable.
Let's solve the equation 14x = 7 for x. Divide both sides by 14:
x = 7/14 = 1/2
We have found the value of x, which is the same as we found using the substitution method. This consistency reinforces our confidence in the solution.
Step 4: Substitute the value back to find the other variable.
Now that we have x = 1/2, we can substitute this value back into either of the original equations to find y. Let's use the first equation, 2x + 3y = 13:
2(1/2) + 3y = 13
1 + 3y = 13
Subtract 1 from both sides:
3y = 12
Divide both sides by 3:
y = 4
We have found the value of y, which is also consistent with our result from the substitution method.
Step 5: Verify the solution.
As before, we should verify that the values x = 1/2 and y = 4 satisfy both original equations. We already did this verification in the substitution method section, and we found that both equations are indeed satisfied.
Result of Elimination Method
Thus, the solution to the system of equations using the elimination method is x = 1/2 and y = 4. This result confirms our previous finding using the substitution method and further validates our solution.
Conclusion
In this article, we successfully solved the system of equations:
- 2x + 3y = 13
- 4x - y = -2
We employed two distinct methods: substitution and elimination. Both methods yielded the same solution: x = 1/2 and y = 4. This solution represents the point (1/2, 4) where the two lines represented by the equations intersect on the coordinate plane.
Understanding and mastering these methods is crucial for solving various mathematical problems and real-world applications. The ability to solve systems of equations provides a powerful tool for modeling and analyzing scenarios where multiple variables are interconnected. Whether you are a student learning algebra or a professional working in a quantitative field, these skills will undoubtedly prove valuable.
We hope this article has provided a clear and comprehensive explanation of how to solve systems of linear equations using substitution and elimination. Remember to practice these techniques with various examples to solidify your understanding and build your problem-solving skills. The more you practice, the more confident and proficient you will become in solving these types of problems.
