How Many Solutions Does The System Of Equations Below Have?${ \begin{array}{l} y = 4x + 2 \ y - 2x = 4 \end{array} }$A. No Solution B. At Least 1 Solution C. Exactly 1 Solution D. More Than 1 Solution

Mar 13, 2025 by ADMIN 205 views

**Solving Systems of Equations: A Comprehensive Guide**

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, economics, and computer science. In this article, we will delve into the world of systems of equations and explore the different types of solutions that a system can have. We will also provide a step-by-step guide on how to solve a system of equations and analyze the results.

What is a System of Equations?

A system of equations is a set of two or more equations that involve variables. Each equation in the system is a statement that two expressions are equal. For example, consider the following system of equations:

{ \begin{array}{l} y = 4x + 2 \\ y - 2x = 4 \end{array} \}

In this system, we have two equations: $y = 4x + 2$ and $y - 2x = 4$ . The first equation is a linear equation in two variables, $x$ and $y$ , while the second equation is also a linear equation in two variables.

Types of Solutions

When solving a system of equations, we can have one of the following types of solutions:

No solution: This occurs when the system is inconsistent, meaning that the equations are contradictory.
At least 1 solution: This occurs when the system has a solution, but we cannot determine the exact number of solutions.
Exactly 1 solution: This occurs when the system has a unique solution, meaning that there is only one possible solution.
More than 1 solution: This occurs when the system has multiple solutions, meaning that there are many possible solutions.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the method of substitution.

Step 1: Solve the First Equation for y

The first equation is $y = 4x + 2$ . We can solve this equation for $y$ by isolating $y$ on one side of the equation.

import sympy as sp
x = sp.symbols('x')
y = sp.symbols('y')

eq1 = sp.Eq(y, 4*x + 2)

y_expr = sp.solve(eq1, y)[0]
print(f"y = {y_expr}")

Step 2: Substitute the Expression for y into the Second Equation

Now that we have an expression for $y$ , we can substitute this expression into the second equation.

# Define the second equation
eq2 = sp.Eq(y - 2*x, 4)

eq2_sub = eq2.subs(y, y_expr)
print(f"{eq2_sub}")

Step 3: Solve the Resulting Equation for x

Now that we have substituted the expression for $y$ into the second equation, we can solve the resulting equation for $x$ .

# Solve the resulting equation for x
x_sol = sp.solve(eq2_sub, x)[0]
print(f"x = {x_sol}")

Step 4: Find the Value of y

Now that we have found the value of $x$ , we can substitute this value into the expression for $y$ to find the value of $y$ .

# Find the value of y
y_sol = y_expr.subs(x, x_sol)
print(f"y = {y_sol}")

Conclusion

In this article, we have explored the concept of systems of equations and the different types of solutions that a system can have. We have also provided a step-by-step guide on how to solve a system of equations using the method of substitution. By following these steps, we can determine the number of solutions that a system of equations has.

Final Answer

Based on the solution to the system of equations, we can determine the final answer.

No solution: The system is inconsistent, meaning that the equations are contradictory.
At least 1 solution: The system has a solution, but we cannot determine the exact number of solutions.
Exactly 1 solution: The system has a unique solution, meaning that there is only one possible solution.
More than 1 solution: The system has multiple solutions, meaning that there are many possible solutions.

In this case, the system of equations has exactly 1 solution.

Code

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

eq1 = sp.Eq(y, 4*x + 2)

y_expr = sp.solve(eq1, y)[0]

eq2 = sp.Eq(y - 2*x, 4)

eq2_sub = eq2.subs(y, y_expr)

x_sol = sp.solve(eq2_sub, x)[0]

y_sol = y_expr.subs(x, x_sol)
print(f"The final answer is {x_sol, y_sol}.")

This code will output the final answer, which is the solution to the system of equations.

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Introduction

Q&A: Solving Systems of Equations

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve variables. Each equation in the system is a statement that two expressions are equal.

Q: What are the different types of solutions that a system of equations can have?

A: When solving a system of equations, we can have one of the following types of solutions:

No solution: This occurs when the system is inconsistent, meaning that the equations are contradictory.
At least 1 solution: This occurs when the system has a solution, but we cannot determine the exact number of solutions.
Exactly 1 solution: This occurs when the system has a unique solution, meaning that there is only one possible solution.
More than 1 solution: This occurs when the system has multiple solutions, meaning that there are many possible solutions.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the method of substitution or elimination. In this case, we will use the method of substitution.

Q: What is the method of substitution?

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

Q: How do I use the method of substitution to solve a system of equations?

A: To use the method of substitution, follow these steps:

Solve one equation for one variable.
Substitute that expression into the other equation.
Solve the resulting equation for the other variable.
Find the value of the first variable by substituting the value of the second variable into one of the original equations.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I use the method of elimination to solve a system of equations?

A: To use the method of elimination, follow these steps:

Multiply both equations by necessary multiples such that the coefficients of one of the variables are the same in both equations.
Add or subtract the equations to eliminate one of the variables.
Solve the resulting equation for the other variable.
Find the value of the first variable by substituting the value of the second variable into one of the original equations.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

Not checking if the system is consistent before solving it.
Not using the correct method of substitution or elimination.
Not solving for both variables in the system.
Not checking if the solution satisfies both equations.

Q: How do I check if a system of equations is consistent?

A: To check if a system of equations is consistent, follow these steps:

Check if the equations are linear.
Check if the equations have the same variables.
Check if the equations have the same coefficients for the variables.
Check if the equations have the same constant terms.

If the equations pass all of these checks, then the system is consistent.

Q: How do I check if a solution satisfies both equations?

A: To check if a solution satisfies both equations, follow these steps:

Substitute the values of the variables into both equations.
Check if the resulting expressions are equal to the constant terms in both equations.

If the resulting expressions are equal to the constant terms in both equations, then the solution satisfies both equations.