Solving Systems Of Equations Determining Solutions For 3x + 2y = 17 And 7x - 5y = 1

Jul 9, 2025 by Admin 84 views

Finding Solutions to Systems of Equations: A Comprehensive Guide

When delving into the world of mathematics, one often encounters systems of equations. These systems, comprising two or more equations with shared variables, require us to find values for these variables that satisfy all equations simultaneously. In this comprehensive guide, we will explore the concept of solutions to systems of equations, focusing on a specific example involving two linear equations. We will dissect the problem, discuss various methods for finding solutions, and provide a step-by-step approach to determine whether a given point is a solution to the system.

Understanding Systems of Equations

A system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is a set of values for the variables that make all equations in the system true simultaneously. Geometrically, each equation in a system represents a curve or a line, and the solution to the system corresponds to the point(s) where these curves or lines intersect. When we talk about solving systems of equations, we're essentially looking for these intersection points.

In the context of linear equations, each equation represents a straight line. A system of two linear equations can have one solution (where the lines intersect at a single point), no solutions (where the lines are parallel and never intersect), or infinitely many solutions (where the lines are coincident, meaning they are the same line). The given problem falls into the category of finding the solution, if it exists, for a system of two linear equations.

The Problem at Hand

We are presented with the following system of linear equations:

3x + 2y = 17
7x - 5y = 1

The question asks us to determine which point, from a given set of options (which are not provided in the initial prompt, but we will address how to approach this generally), is a solution to this system. A point, in this context, is an ordered pair (x, y) that represents a location on a coordinate plane. To check if a point is a solution, we substitute the x and y values of the point into both equations. If the point satisfies both equations, then it is a solution to the system.

Methods for Solving Systems of Equations

Before we dive into verifying potential solutions, let's briefly discuss some common methods for solving systems of equations. Understanding these methods provides a broader context for the problem and can be useful in cases where we need to find the solution ourselves, rather than just checking given options.

1. Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which can then be solved. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

2. Elimination Method

The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. Then, we add the equations together, which eliminates one variable and leaves us with a single equation with one variable. This equation can be solved, and the value can be substituted back into one of the original equations to find the other variable.

3. Graphical Method

The graphical method involves graphing both equations on the same coordinate plane. The solution to the system is the point(s) where the graphs intersect. This method is particularly useful for visualizing the system and understanding the nature of the solutions, but it may not be the most accurate method for finding precise solutions.

4. Matrix Methods

For larger systems of equations, matrix methods, such as Gaussian elimination or matrix inversion, can be more efficient. These methods involve representing the system of equations in matrix form and then using matrix operations to solve for the variables.

Verifying a Solution

Now, let's focus on the core task: verifying whether a given point is a solution to the system.

The general approach is as follows:

Obtain the point: We need a specific point (x, y) to test. Let's assume, for the sake of illustration, that we want to test the point (3, 4).
Substitute the values: Substitute the x and y values of the point into both equations in the system.
- For the first equation (3x + 2y = 17), substitute x = 3 and y = 4:
```
3(3) + 2(4) = 9 + 8 = 17
```
  The equation holds true.
- For the second equation (7x - 5y = 1), substitute x = 3 and y = 4:
```
7(3) - 5(4) = 21 - 20 = 1
```
  The equation holds true.
Check if both equations are satisfied: If the point satisfies both equations, then it is a solution to the system. In our example, the point (3, 4) satisfies both equations, so it is a solution.
If either equation is not satisfied: If the point does not satisfy at least one of the equations, then it is not a solution to the system.

Step-by-Step Solution to the Given System

Let's solve the given system of equations using the elimination method to find the actual solution. This will give us a concrete answer to compare against if we were given multiple-choice options.

3x + 2y = 17  (Equation 1)
7x - 5y = 1   (Equation 2)

Multiply to eliminate a variable: We can eliminate 'y' by multiplying Equation 1 by 5 and Equation 2 by 2:
- Equation 1 * 5: 15x + 10y = 85
- Equation 2 * 2: 14x - 10y = 2
Add the equations: Adding the modified equations eliminates 'y':
```
(15x + 10y) + (14x - 10y) = 85 + 2
29x = 87
```
Solve for x: Divide both sides by 29:
```
x = 87 / 29
x = 3
```
Substitute x back into an equation: Substitute x = 3 into Equation 1:
```
3(3) + 2y = 17
9 + 2y = 17
2y = 8
```
Solve for y: Divide both sides by 2:
```
y = 8 / 2
y = 4
```

Therefore, the solution to the system is (3, 4). This confirms our earlier test case where we assumed the point (3,4) and verified it as a solution.

Key Considerations and Potential Pitfalls

Careful Substitution: When substituting values, it's crucial to pay close attention to signs and perform the arithmetic correctly. A small error in substitution can lead to an incorrect conclusion.
Checking Both Equations: Remember that a solution must satisfy all equations in the system. If a point satisfies one equation but not the other, it is not a solution to the system.
No Solution or Infinite Solutions: Some systems may have no solution (parallel lines) or infinitely many solutions (coincident lines). When verifying points, you might encounter situations where no point satisfies both equations or where any point on a line seems to work.
Fractions and Decimals: Solutions may not always be integers. Be prepared to work with fractions or decimals when solving systems of equations.

Conclusion

Determining whether a point is a solution to a system of equations is a fundamental skill in algebra. By substituting the coordinates of the point into each equation and verifying that all equations are satisfied, we can confidently identify solutions. Understanding the methods for solving systems of equations, such as substitution, elimination, and graphing, provides a broader understanding of the problem and can be helpful in finding solutions when they are not explicitly provided. In the given example, we not only demonstrated the verification process but also solved the system to confirm that (3, 4) is indeed the solution. When faced with similar problems, remember to be meticulous in your substitutions and calculations, and always check that the point satisfies all equations in the system.

This comprehensive guide has equipped you with the knowledge and tools to confidently tackle problems involving systems of equations and their solutions. Practice applying these methods to various examples, and you'll become proficient in solving these mathematical puzzles.