Which Equation is the Inverse of 5y + 4 = (x + 3)² + 1/2?

In mathematics, finding the inverse of an equation is a fundamental concept. An inverse equation, when applied to another equation, undoes the operations performed by that equation. This article explores the process of determining the inverse equation of the given quadratic equation: 5y + 4 = (x + 3)² + ½. By following a step-by-step approach, we will decipher the inverse equation and discuss its properties.

I. Understanding the Given Equation:

Before diving into the inverse equation, let's comprehend the given quadratic equation: 5y + 4 = (x + 3)² + ½. This equation represents a parabola in the vertex form. The vertex form of a quadratic equation is given by y = a(x - h)² + k, where (h, k) represents the vertex coordinates.

The Given Equation

II. Step 1: Isolate the Quadratic Term:

To find the inverse equation, we need to isolate the quadratic term. By subtracting 4 and then dividing by 5, we can rewrite the equation as follows: y = [(x + 3)² + ½ - 4]/5.

III. Step 2: Swap x and y:

To transform the equation into its inverse form, we interchange x and y. After this substitution, the equation becomes x = [(y + 3)² + ½ - 4]/5.

IV. Step 3: Solve for y:

Now, let's solve the equation for y, which will give us the inverse equation. Begin by multiplying both sides of the equation by 5 to eliminate the fraction: 5x = (y + 3)² + ½ - 4.

Next, simplify the equation by combining like terms: 5x = (y + 3)² - 7/2.

To eliminate the squared term, take the square root of both sides: √(5x) = √[(y + 3)² - 7/2].

Squaring both sides to remove the radical sign, we get: 5x = (y + 3)² - 7/2.

Solve for y

V. Step 4: Simplify the Equation:

To further simplify the equation, expand the square term on the right side: 5x = y² + 6y + 9 - 7/2.

Combining like terms and converting the fractions into a common denominator: 10x = 2y² + 12y + 18 - 7.

Simplifying the equation: 10x = 2y² + 12y + 11.

VI. Step 5: Rearrange the Equation:

To obtain the inverse equation in standard form, rearrange the terms as follows: 2y² + 12y + 11 - 10x = 0.

VII. Conclusion:

After carefully following the step-by-step process, we have determined the inverse equation of 5y + 4 = (x + 3)² + ½ to be 2y² + 12y + 11 - 10x = 0. The inverse equation undoes the operations performed by the given quadratic equation, allowing us to find the original values of x and y from their transformed counterparts.

Understanding the inverse equation provides valuable insights into the relationships between variables in the original equation. By employing the concepts discussed in this article, one can confidently explore the properties and characteristics of inverse equations in mathematics.