Solving Systems Of Equations A Comprehensive Guide

Jul 13, 2025 by ADMIN 51 views

$\begin{array}{l} 2 x-y=7 \\ y=2 x+3 \end{array}$

A. $(2,3)$ B. $(2,7)$ C. no solution D. infinite number of solutions

In mathematics, solving a system of equations is a fundamental skill. This article delves into the process of finding the solution to the given system of equations:

$\begin{array}{l} 2 x-y=7 \\ y=2 x+3 \end{array}$

We'll explore different methods and explain why one particular answer from the provided options (A. $(2,3)$ , B. $(2,7)$ , C. no solution, D. infinite number of solutions) is the correct one. Understanding how to solve systems of equations is crucial not only for academic success but also for various real-world applications.

Methods for Solving Systems of Equations

There are several methods available to solve systems of equations, each with its own strengths and weaknesses. We will focus on two primary methods here: substitution and elimination. These methods allow us to manipulate the equations in a way that isolates the variables and leads us to the solution.

1. Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily 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. This method is particularly useful when one of the equations is already solved for one variable, as is the case in our given system.

In our system,

$\begin{array}{l} 2 x-y=7 \\ y=2 x+3 \end{array}$

the second equation, $y = 2x + 3$ , is already solved for $y$ . We can directly substitute this expression for $y$ into the first equation.

2. Elimination Method

The elimination method (also known as the addition method) involves manipulating the equations so that the coefficients of one of the variables are opposites. When the equations are added together, that variable is eliminated, leaving a single equation with one variable. This method is particularly useful when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant. After finding the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

To use the elimination method for our system,

$\begin{array}{l} 2 x-y=7 \\ y=2 x+3 \end{array}$

we can rearrange the second equation to have the same form as the first equation. Then, we can see if the coefficients allow for easy elimination.

Solving the System Using Substitution

Let's apply the substitution method to solve the given system of equations. As mentioned earlier, the second equation, $y = 2x + 3$ , is already solved for $y$ . We will substitute this expression into the first equation:

$2x - y = 7$

Substitute $y$ with $2x + 3$ :

$2x - (2x + 3) = 7$

Now, we simplify and solve for $x$ :

$2x - 2x - 3 = 7$

$-3 = 7$

This equation, $-3 = 7$ , is a contradiction. This means there is no value of $x$ that can make this equation true. Therefore, the system of equations has no solution.

Detailed Steps for Substitution

Identify the equations:

$\begin{array}{l} 2 x-y=7 ext{ (Equation 1)} \\ y=2 x+3 ext{ (Equation 2)} \end{array}$
Substitute Equation 2 into Equation 1:

$2x - (2x + 3) = 7$
Distribute the negative sign:

$2x - 2x - 3 = 7$
Simplify the equation:

$-3 = 7$
Interpret the result:

Since $-3 = 7$ is a false statement, there is no solution to the system of equations.

Verifying the Solution Graphically

Another way to understand why this system has no solution is to consider the graphical representation of the equations. Each equation represents a line in the coordinate plane. The solution to the system is the point where the lines intersect. If the lines are parallel, they will never intersect, indicating that there is no solution.

Let's rewrite the equations in slope-intercept form ( $y = mx + b$ ) to easily identify their slopes and y-intercepts.

Equation 1: $2x - y = 7$ can be rewritten as $y = 2x - 7$ .

Equation 2: $y = 2x + 3$ is already in slope-intercept form.

We can see that both equations have the same slope ( $m = 2$ ) but different y-intercepts. This means the lines are parallel and will never intersect. Therefore, there is no solution to the system of equations.

Graphical Representation

Equation 1: $y = 2x - 7$ (Slope = 2, y-intercept = -7)
Equation 2: $y = 2x + 3$ (Slope = 2, y-intercept = 3)

Since the slopes are the same and the y-intercepts are different, the lines are parallel and do not intersect.

Why Other Options Are Incorrect

Let's briefly discuss why the other options are incorrect:

A. (2, 3): If we substitute $x = 2$ $x = 2$ and $y = 3$ $y = 3$ into the equations:
- $2(2) - 3 = 4 - 3 = 1 eq 7$ (Fails Equation 1)
- $3 = 2(2) + 3 = 4 + 3 = 7$ (Fails Equation 2)
B. (2, 7): If we substitute $x = 2$ $x = 2$ and $y = 7$ $y = 7$ into the equations:
- $2(2) - 7 = 4 - 7 = -3 eq 7$ (Fails Equation 1)
- $7 = 2(2) + 3 = 4 + 3 = 7$ (Satisfies Equation 2)
D. Infinite number of solutions: This would be the case if the two equations represented the same line. However, since the lines have different y-intercepts, they are distinct parallel lines.

Conclusion

In conclusion, when we solve the system of equations

$\begin{array}{l} 2 x-y=7 \\ y=2 x+3 \end{array}$

using the substitution method, we arrive at a contradiction ( $-3 = 7$ ). This indicates that there is no solution to the system. Furthermore, by analyzing the equations in slope-intercept form and recognizing that they represent parallel lines, we can confirm this result graphically. Understanding the different methods for solving systems of equations and being able to interpret the results is essential for mathematical proficiency. Therefore, the correct answer is C. no solution.

This comprehensive exploration highlights the importance of carefully applying algebraic techniques and graphical analysis to accurately solve systems of equations. Remember to always verify your solutions and consider the geometric interpretation to deepen your understanding.