Problem 9 - Algebra and Trigonometry

Question

\(5-10\) . Determine whether the given points are on the graph of the equation. $$ x^{2}+x y+y^{2}=4 ; \quad(0,-2),(1,-2),(2,-2) $$

Step-by-Step Solution

Answer

Points \((0, -2)\) and \((2, -2)\) are on the graph; \((1, -2)\) is not.

Step 1: Substitute the First Point

We need to determine if the point \((0, -2)\) satisfies the given equation \(x^2 + xy + y^2 = 4\). Substitute \(x = 0\) and \(y = -2\) into the equation:\[0^2 + 0(-2) + (-2)^2 = 4\]

Step 2: Simplify the Equation

Simplify the right-hand side of the equation after substitution: \[0 + 0 + 4 = 4\]Since both sides of the equation are equal, \((0, -2)\) is on the graph.

Step 3: Substitute the Second Point

Now check the second point \((1, -2)\) by substituting \(x = 1\) and \(y = -2\) into the equation:\[1^2 + 1(-2) + (-2)^2 = 4\]

Step 4: Simplify the Second Point's Equation

Simplify:\[1 - 2 + 4 = 3\]Here, the left-hand side does not equal the right-hand side (4), so \((1, -2)\) is not on the graph.

Step 5: Substitute the Third Point

Finally, check the third point \((2, -2)\) by substituting \(x = 2\) and \(y = -2\) into the equation:\[2^2 + 2(-2) + (-2)^2 = 4\]

Step 6: Simplify the Third Point's Equation

Simplify:\[4 - 4 + 4 = 4\]The equation holds true, so \((2, -2)\) is on the graph.

Key Concepts

Graph of an equation

A graph of an equation is a visual representation of all the solutions to that equation on a coordinate plane. Each solution or point on the graph satisfies the equation. For example, when we have an equation like \(x^2 + xy + y^2 = 4\), we are looking for all the coordinate pairs \((x, y)\) that make the equation true.

When graphed, such an equation typically forms a shape based on its degree and terms. Quadratic equations, like our example, often form ellipses, parabolas, or hyperbolas. The graph helps us to see all possible solutions at a glance, and it can be a powerful tool for understanding complex relationships between variables.

• The points that satisfy the equation are called solutions.
• The shape of the graph depends on the equation's form and degree.
• Graphs allow us to visualize and better understand mathematical relationships.

Understanding the graph of an equation in coordinate geometry allows us to see how changes in one variable affect another. It is a foundational concept that aids in visualizing solutions in algebra and calculus.

Checking points

Checking points on the graph of an equation involves determining if specific coordinate pairs satisfy the given equation. This is an essential part of coordinate geometry, where you verify whether a point is actually on the graph.

To check if a point \((x, y)\) is on the graph of an equation, simply substitute the x and y values into the equation. If the left-hand side equals the right-hand side when simplified, the point is on the graph. In our example with the equation \(x^2 + xy + y^2 = 4\):

• For \((0, -2)\), substituting gives \(0^2 + 0(-2) + (-2)^2 = 4\), which simplifies correctly, indicating the point is on the graph.
• For \((1, -2)\), we get \(1^2 + 1(-2) + (-2)^2 = 3\), which isn't equal to 4, so the point isn't on the graph.
• For \((2, -2)\), substituting gives \(2^2 + 2(-2) + (-2)^2 = 4\), which simplifies correctly, confirming it's on the graph.

Checking points helps in identifying whether specific solutions fit into the equation, ensuring accuracy when plotting graphs or solving problems. It is a straightforward procedure that requires basic substitution and calculation.

Substitution method

The substitution method is a technique used to find out if a point lies on the graph by replacing the variables in an equation with specific values. In coordinate geometry, this involves plugging in the x and y values of a point into the equation to see if it holds true.

This method is straightforward: take each point and plug its x and y values into the equation. If the equation balances (left side equals right side), the point is part of the graph.

• Step 1: Identify the point's coordinates. Example: \((x, y) = (0, -2)\).
• Step 2: Substitute the coordinates into the equation: \(x^2 + xy + y^2 = 4\).
• Step 3: Simplify the equation and check if both sides are equal.

This method is crucial for graphing and problem-solving in mathematics. By using substitution effectively, one can quickly verify potential solutions to an equation or determine if a given point is on the intended curve or line. It is a key technique that simplifies understanding and checking mathematical tasks in coordinate geometry.