Solving the System of Equations: $y - x + 3 = 0$ and $x^2 - 3xy + y^2 + 19 = 0$

Introduction

Key Concepts

Understanding Simultaneous Equations

Simultaneous equations involve finding the values of variables that satisfy multiple equations at the same time. In the context of the Cambridge IGCSE Mathematics - Additional syllabus, students are expected to solve both linear and non-linear systems. The given system consists of one linear equation and one quadratic equation, which introduces additional complexity to the solution process.

Analyzing the Given Equations

The system provided is:

• Equation 1: $y - x + 3 = 0$
• Equation 2: $x^2 - 3xy + y^2 + 19 = 0$

Equation 1 is linear in nature, while Equation 2 is quadratic. The presence of quadratic terms indicates that the system might have multiple solutions, including real and complex roots.

Method of Substitution

The substitution method is an effective technique for solving systems where one equation can be easily rearranged to express one variable in terms of another. Here’s how to apply it:

1. Rearrange Equation 1 to express $y$ in terms of $x$: $$ y = x - 3 $$
2. Substitute $y = x - 3$ into Equation 2: $$ x^2 - 3x(x - 3) + (x - 3)^2 + 19 = 0 $$
3. Simplify the equation: $$ x^2 - 3x^2 + 9x + x^2 - 6x + 9 + 19 = 0 $$ $$ (-x^2 + 3x + 28) = 0 $$ $$ x^2 - 3x - 28 = 0 $$
4. Solve the quadratic equation: $$ x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-28)}}{2(1)} $$ $$ x = \frac{3 \pm \sqrt{9 + 112}}{2} $$ $$ x = \frac{3 \pm \sqrt{121}}{2} $$ $$ x = \frac{3 \pm 11}{2} $$ $$ x = 7 \quad \text{or} \quad x = -4 $$
5. Find corresponding $y$ values:
   • For $x = 7$: $$ y = 7 - 3 = 4 $$
   • For $x = -4$: $$ y = -4 - 3 = -7 $$

Thus, the solutions are $(7, 4)$ and $(-4, -7)$.

Graphical Interpretation

Graphing the two equations provides a visual representation of the solutions. The linear equation $y = x - 3$ is a straight line with a slope of 1 and a y-intercept at -3. The quadratic equation $x^2 - 3xy + y^2 + 19 = 0$ represents a conic section, and its graph intersects the linear equation at the points representing the solutions found earlier.

Verification of Solutions

It is crucial to verify the solutions by substituting them back into the original equations:

• For $(7, 4)$:
  • Equation 1: $$ 4 - 7 + 3 = 0 $$ $$ 0 = 0 \quad \text{(True)} $$
  • Equation 2: $$ 7^2 - 3(7)(4) + 4^2 + 19 = 0 $$ $$ 49 - 84 + 16 + 19 = 0 $$ $$ 0 = 0 \quad \text{(True)} $$
• For $(-4, -7)$:
  • Equation 1: $$ -7 - (-4) + 3 = 0 $$ $$ -7 + 4 + 3 = 0 $$ $$ 0 = 0 \quad \text{(True)} $$
  • Equation 2: $$ (-4)^2 - 3(-4)(-7) + (-7)^2 + 19 = 0 $$ $$ 16 - 84 + 49 + 19 = 0 $$ $$ 0 = 0 \quad \text{(True)} $$

Both solutions satisfy the original system of equations, confirming their validity.

Alternative Methods

Besides substitution, other methods such as elimination or graphical methods can be employed to solve systems of equations. However, in this particular case, substitution provides a straightforward pathway to the solution.

Advanced Concepts

Exploring Quadratic Systems

Quadratic systems, like the one discussed, present unique challenges compared to purely linear systems. The presence of squared terms requires a deeper understanding of algebraic manipulation and equation solving techniques.

Discriminant Analysis

The discriminant of a quadratic equation, given by $D = b^2 - 4ac$, plays a pivotal role in determining the nature of the roots. In the substitution step, the quadratic equation $x^2 - 3x - 28 = 0$ has:

• $a = 1$
• $b = -3$
• $c = -28$

Thus, the discriminant is:

A positive discriminant indicates two distinct real roots, which aligns with the two solutions obtained.

Systems with Multiple Solutions

In systems involving quadratic equations, the number of solutions can vary based on the nature of the equations. Here, the system intersects at two distinct points, but depending on the coefficients and constants, systems can have one, two, or no real solutions. Additionally, some systems may yield complex solutions.

Parametric Solutions

For more complex systems, introducing parameters can simplify the solving process. By expressing one variable in terms of another using a parameter, the system can sometimes be reduced to a single equation in one variable, facilitating easier solutions.

Applications of Simultaneous Equations

Simultaneous equations are not just academic; they have practical applications in various fields:

• Economics: Determining equilibrium points where supply equals demand.
• Engineering: Solving for currents in electrical circuits using Kirchhoff's laws.
• Physics: Calculating forces in statics by balancing multiple equations.
• Computer Science: Optimizing algorithms by solving system constraints.

Integration with Other Mathematical Concepts

Understanding simultaneous equations lays the foundation for more advanced topics such as linear algebra, differential equations, and optimization problems. Mastery of these concepts is essential for higher studies in mathematics, engineering, economics, and the sciences.

Challenging Problems and Solutions

To deepen understanding, consider the following complex problem:

Problem: Solve the system of equations: $$ y - 2x + 5 = 0 $$ $$ 2x^2 - 4xy + y^2 + 25 = 0 $$

Solution:

1. Express $y$ from the first equation: $$ y = 2x - 5 $$
2. Substitute into the second equation: $$ 2x^2 - 4x(2x - 5) + (2x - 5)^2 + 25 = 0 $$
3. Simplify: $$ 2x^2 - 8x^2 + 20x + 4x^2 - 20x + 25 + 25 = 0 $$ $$ (-2x^2 + 0x + 50) = 0 $$ $$ 2x^2 = 50 $$ $$ x^2 = 25 $$ $$ x = 5 \quad \text{or} \quad x = -5 $$
4. Find corresponding $y$ values:
   • For $x = 5$: $$ y = 2(5) - 5 = 5 $$
   • For $x = -5$: $$ y = 2(-5) - 5 = -15 $$

Thus, the solutions are $(5, 5)$ and $(-5, -15)$. Verification confirms their validity.

Complex Numbers in Systems of Equations

Some systems may yield complex solutions, especially when dealing with quadratic equations. Understanding complex numbers and their properties is essential for solving such systems. For example, the equation $x^2 + 1 = 0$ has solutions $x = i$ and $x = -i$, where $i$ is the imaginary unit.

Matrix Representation and Solving

While matrices are typically used for linear systems, they can also assist in solving non-linear systems through linearization techniques. Representing equations in matrix form can facilitate the use of computational tools and linear algebra methods.

Comparison Table

Summary and Key Takeaways