Unlocking the Equation- Discovering the Solution to the Given System of Equations

What is the solution to the following system of equations?

In mathematics, solving systems of equations is a fundamental skill that is essential for understanding more complex mathematical concepts. Whether it’s in algebra, calculus, or even in real-world applications, the ability to find the solution to a system of equations is crucial. In this article, we will explore different methods to solve a system of equations and provide a step-by-step guide to finding the solution. Let’s dive into the world of equations and discover how to unravel the mystery behind “what is the solution to the following system of equations?”

First, let’s take a look at the system of equations we are going to solve:

Equation 1: 2x + 3y = 7
Equation 2: 4x – y = 5

To find the solution, we can use various methods such as substitution, elimination, or matrix algebra. In this article, we will focus on the substitution and elimination methods, as they are the most commonly used techniques for solving systems of linear equations.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This allows us to solve for the remaining variable and, subsequently, find the value of the other variable.

1. Solve Equation 1 for x: 2x = 7 – 3y
2. Divide both sides by 2: x = (7 – 3y) / 2
3. Substitute x into Equation 2: 4((7 – 3y) / 2) – y = 5
4. Simplify the equation: 2(7 – 3y) – y = 5
5. Distribute the 2: 14 – 6y – y = 5
6. Combine like terms: -7y = -9
7. Solve for y: y = 9 / 7
8. Substitute y back into Equation 1: 2x + 3(9 / 7) = 7
9. Simplify the equation: 2x + 27 / 7 = 7
10. Multiply both sides by 7: 14x + 27 = 49
11. Subtract 27 from both sides: 14x = 22
12. Solve for x: x = 22 / 14
13. Simplify the fraction: x = 11 / 7

The solution to the system of equations is x = 11 / 7 and y = 9 / 7.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, allowing us to solve for the remaining variable. Once we have the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.

1. Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of y equal: 4x + 6y = 14 and 12x – 3y = 15
2. Add the two equations: 16x + 3y = 29
3. Solve for x: 16x = 29 – 3y
4. Divide both sides by 16: x = (29 – 3y) / 16
5. Substitute x into Equation 1: 2((29 – 3y) / 16) + 3y = 7
6. Simplify the equation: (29 – 3y) / 8 + 3y = 7
7. Multiply both sides by 8: 29 – 3y + 24y = 56
8. Combine like terms: 21y = 27
9. Solve for y: y = 27 / 21
10. Simplify the fraction: y = 9 / 7
11. Substitute y back into Equation 1: 2x + 3(9 / 7) = 7
12. Simplify the equation: 2x + 27 / 7 = 7
13. Multiply both sides by 7: 14x + 27 = 49
14. Subtract 27 from both sides: 14x = 22
15. Solve for x: x = 22 / 14
16. Simplify the fraction: x = 11 / 7

The solution to the system of equations is x = 11 / 7 and y = 9 / 7.

In conclusion, solving a system of equations requires patience and practice. By using the substitution or elimination method, we can find the solution to the given system of equations. Remember, the key is to be systematic and pay close attention to the steps involved in each method. With time and practice, you will become more proficient in solving systems of equations and applying this skill to various mathematical problems.