Solve The Following System Of Equations:${ \begin{array}{l} x^2 - Xy = 0 \ x + Y = 1 \end{array} }$

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two equations, where one equation is quadratic and the other is linear. We will use algebraic methods to solve the system and provide a step-by-step approach to finding the solutions.

The System of Equations

The system of equations we will be solving is given by:

$$\begin{array}{l} x^2 - xy = 0 \\ x + y = 1 \end{array}$$

Step 1: Analyze the First Equation

The first equation is a quadratic equation in terms of $x$ and $y$. We can rewrite it as:

$$x^2 - xy = 0$$

This equation can be factored as:

$$x(x - y) = 0$$

From this factorization, we can see that either $x = 0$ or $x - y = 0$.

Step 2: Analyze the Second Equation

The second equation is a linear equation in terms of $x$ and $y$. We can rewrite it as:

$$x + y = 1$$

This equation can be solved for $y$ as:

$$y = 1 - x$$

Step 3: Substitute the Expression for $y$ into the First Equation

We can substitute the expression for $y$ into the first equation as:

$$x(x - (1 - x)) = 0$$

Simplifying this expression, we get:

$$x(x - 1 + x) = 0$$

$$x(2x - 1) = 0$$

Step 4: Solve for $x$

We can solve for $x$ by setting each factor equal to zero:

$$x = 0 \quad \text{or} \quad 2x - 1 = 0$$

Solving for $x$ in the second equation, we get:

$$2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2}$$

Step 5: Find the Corresponding Values of $y$

We can find the corresponding values of $y$ by substituting the values of $x$ into the expression for $y$:

$$y = 1 - x$$

For $x = 0$, we get:

$$y = 1 - 0 = 1$$

For $x = \frac{1}{2}$, we get:

$$y = 1 - \frac{1}{2} = \frac{1}{2}$$

Conclusion

In this article, we solved a system of two equations using algebraic methods. We analyzed the first equation, which was a quadratic equation, and factored it to find the possible values of $x$. We then substituted the expression for $y$ into the first equation and solved for $x$. Finally, we found the corresponding values of $y$ by substituting the values of $x$ into the expression for $y$. The solutions to the system of equations are $x = 0, y = 1$ and $x = \frac{1}{2}, y = \frac{1}{2}$.

Final Answer

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Introduction

In our previous article, we solved a system of two equations using algebraic methods. In this article, we will provide a Q&A guide to help you understand the concepts and methods used to solve the system of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. In our previous article, we solved a system of two equations, where one equation was quadratic and the other was linear.

Q: What are the different methods used to solve a system of equations?

A: There are several methods used to solve a system of equations, including:

• Substitution method: This method involves substituting the expression for one variable into the other equation.
• Elimination method: This method involves eliminating one variable by adding or subtracting the equations.
• Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting the expression for one variable into the other equation. For example, if we have the equations:

$$x + y = 1$$

$$x^2 - xy = 0$$

We can substitute the expression for $y$ into the second equation as:

$$x^2 - x(1 - x) = 0$$

Simplifying this expression, we get:

$$x^2 - x + x^2 = 0$$

$$2x^2 - x = 0$$

Q: What is the elimination method?

A: The elimination method involves eliminating one variable by adding or subtracting the equations. For example, if we have the equations:

$$x + y = 1$$

$$x^2 - xy = 0$$

We can multiply the first equation by $x$ and add it to the second equation as:

$$x^2 + xy = x$$

$$x^2 - xy = 0$$

Adding these two equations, we get:

$$2x^2 = x$$

Simplifying this expression, we get:

$$2x^2 - x = 0$$

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. For example, if we have the equations:

$$x + y = 1$$

$$x^2 - xy = 0$$

We can graph these equations on a coordinate plane and find the point of intersection.

Q: How do I know which method to use?

A: The choice of method depends on the type of equations and the variables involved. If the equations are linear, the substitution or elimination method may be used. If the equations are quadratic, the substitution or elimination method may be used. If the equations are complex, the graphical method may be used.

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

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

• Not checking the solutions: Make sure to check the solutions to ensure that they satisfy both equations.
• Not using the correct method: Choose the correct method based on the type of equations and variables involved.
• Not simplifying the expressions: Simplify the expressions to make it easier to solve the system of equations.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and methods used to solve a system of equations. We discussed the substitution, elimination, and graphical methods, and provided examples to illustrate each method. We also discussed common mistakes to avoid when solving a system of equations. By following these guidelines, you can become proficient in solving systems of equations and apply this knowledge to a wide range of mathematical and real-world problems.

Final Answer

The final answer is $\boxed{\left( 0, \ 1\right), \left( \frac{1}{2}, \ \frac{1}{2}\right)}$.