Problem 36 - Contemporary Precalculus

Question

Determine whether the point is on the graph of the given equation. $$(2,-1) ; x^{2}+y^{2}-6 x+8 y=-15$$

Step-by-Step Solution

Answer

Answer: Yes, the point (2, -1) lies on the graph of the equation.

Step 1: Substitute coordinates

We need to substitute the coordinates of the given point \((2, -1)\) into the equation \(x^2 + y^2 - 6x + 8y = -15\). So, replace \(x\) by \(2\) and \(y\) by \(-1\) and simplify the equation to see if it holds true.

Step 2: Simplify and check if the equation holds true

Now, we will simplify the resulting expression after substituting the coordinates: $$(2)^2 + (-1)^2 - 6(2) + 8(-1) = -15$$ $$4 + 1 - 12 - 8 \stackrel{?}{=} -15$$ $$-7 - 8 \stackrel{?}{=} -15$$ $$-15 = -15$$ It holds true, so the point \((2, -1)\) lies on the graph of the given equation.

Key Concepts

Coordinate Substitution

When working with algebraic equations, particularly when graphing, coordinate substitution is a valuable technique. It allows you to verify if a certain point lies on the graph of an equation. To perform coordinate substitution, you take the x and y values of your point and replace the x and y variables in the equation with these numbers, respectively.

For our exercise, the given point is (2, -1), which means we substitute 2 for x and -1 for y in the equation. This approach tests the validity of the point in relation to the equation. If, after substitution, the equation balances or is 'true,' the point is part of the graph for that equation. Coordinate substitution can be applied to any equation, including, but not limited to, linear, quadratic, or any other polynomial equation.

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is two. The standard form of a quadratic equation is (ax^2 + bx + c = 0), where a, b, and c are constants, and 'a' is not zero. The graph of a quadratic equation is known as a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'.

In the example given, (x^2 + y^2 - 6x + 8y +15 = 0), the equation represents a circle, which is a special case of a quadratic equation because it includes both x^2 and y^2 terms. Knowing the general shape of the graph can aid significantly in visualization and understanding the relationship between the equation and the points that lie on its graph.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with and understand. This process may include combining like terms, applying the distributive property, and following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In the problem provided, simplifying the expression after substituting the coordinates (2 for x and -1 for y) was crucial. We first calculated the squared terms, then the products involving 6 and 2, and finally the products involving 8 and -1. The addition and subtraction of these results gave us -15, which matched the right-hand side of the original equation. The ability to simplify expressions accurately ensures that you can correctly assess the validity of points on graphs and solve equations efficiently.