Problem 56 - Algebra 2

Question

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ -x y-2 x-3 y+6=0 $$

Step-by-Step Solution

Answer

A = 0, B = -1, C = 0

Step 1: Identify the coefficients

The given equation is \[-xy-2x-3y+6=0\]. To match the standard form \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\), identify the coefficients of each term.

Step 2: Identify A

In the given equation, there is no \(x^{2}\) term, which means the coefficient \(A\) for \(x^{2}\) is 0.

Step 3: Identify B

The term involving both \(x\) and \(y\) is \(-xy\). Therefore, the coefficient \(B\) is \(-1\).

Step 4: Identify C

There is no \(y^{2}\) term in the equation, which means the coefficient \(C\) for \(y^{2}\) is 0.

Key Concepts

Coefficients in Standard Form

When dealing with quadratic equations, placing them in a standard form is essential for problem-solving. A standard form for these equations is expressed as: \[ Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 \] Here, each letter represents a coefficient, a constant number that quantifies the relation between terms in the equation. The letters x and y are variables that stand in for numbers we might solve for.

• A is the coefficient of the \(x^{2}\) term
• B is the coefficient of the \(xy\) term
• C is the coefficient of the \(y^{2}\) term
• D is the coefficient of the \(x\) term
• E is the coefficient of the \(y\) term
• F is a constant term

Recognizing these coefficients correctly is a critical step, as it helps in understanding the behavior of the equation. For example, determining that a certain coefficient is zero, as is the case with \(A\) and \(C\) in our given equation, simplifies the equation and immediately points out the absence of those squared terms.

Identifying Coefficients

Identifying coefficients in an equation, especially one in the standard form, is like finding the right pieces to solve a puzzle. Consider the equation \[-xy - 2x - 3y + 6 = 0\].This equation can be quite different in appearance from the standard form, so it requires careful assessment.First, look at each part of the equation:

• For a term with both x and y, like \(-xy\), B is the coefficient. Here, B equals \(-1\).
• If there is no \(x^2\) or \(y^2\) present, A and C are both 0, indicating those terms are not part of the equation.

By evaluating the equation piece by piece not only do you simplify it, but you make it easier to manipulate. This is especially handy in scenarios where equations might need further adjustments or derivations. Identifying each coefficient is the first step toward transforming or solving the equation.

Polynomial Equations

Polynomial equations like the ones we are tackling come in various forms and order based on the degree of the polynomial's highest power. For quadratic equations, such as our example, the highest power of either variable is two, as seen in a term like \(Ax^{2}\) or \(Cy^{2}\). Though squared terms are not included in our specific equation, understanding their role is crucial.

• Quadratics can represent parabolas when graphed, and their coefficients (like A and C) impact the shape and direction.
• In fact, variations in coefficients help determine intersections, vertex positions, and axes of symmetry in geometric interpretations.

The importance of mastering quadratic polynomial equations lies in their application to real-world problems, from physics to economics. They can describe integer sequences, control trajectories, and model diverse phenomena. Recognizing how coefficients reflect changes in these equations forms the basis for deeper mathematical understanding and application.