Find Y To The Nearest Hundredths Place

Finding 'y' to the Nearest Hundredth: A Comprehensive Guide

Finding the value of 'y' to the nearest hundredth involves several mathematical techniques, depending on the context in which 'y' is presented. This guide will explore various scenarios, offering step-by-step solutions and practical examples to help you master this skill. We'll cover everything from simple equations to more complex scenarios involving trigonometric functions and calculus. Understanding the nuances of each method is crucial for accurate and efficient problem-solving.

Solving for 'y' in Linear Equations

The simplest scenario involves solving for 'y' in a linear equation. A linear equation is an equation of a straight line, typically expressed in the form ax + by = c, where 'a', 'b', and 'c' are constants.

Example 1: Solve for 'y' in the equation 2x + 3y = 12.

To solve for 'y', we need to isolate 'y' on one side of the equation:

1. Subtract 2x from both sides: 3y = 12 - 2x
2. Divide both sides by 3: y = (12 - 2x) / 3

Now, let's assume we need to find the value of 'y' when x = 2. Substitute x = 2 into the equation:

y = (12 - 2(2)) / 3 = (12 - 4) / 3 = 8 / 3 ≈ 2.67

Therefore, to the nearest hundredth, y ≈ 2.67.

Example 2: Solve for 'y' in the equation -5x + y = 7 when x = 1.5

1. Add 5x to both sides: y = 7 + 5x
2. Substitute x = 1.5: y = 7 + 5(1.5) = 7 + 7.5 = 14.5

In this case, y = 14.50 to the nearest hundredth.

Solving for 'y' in Quadratic Equations

Quadratic equations are equations of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and a ≠ 0. Solving for 'y' (assuming the equation is written as a function of y, such as ay² + by + c = 0) requires the use of the quadratic formula:

y = (-b ± √(b² - 4ac)) / 2a

Example 3: Solve for 'y' in the equation y² - 4y + 3 = 0.

Here, a = 1, b = -4, and c = 3. Substituting these values into the quadratic formula:

y = (4 ± √((-4)² - 4 * 1 * 3)) / (2 * 1) y = (4 ± √(16 - 12)) / 2 y = (4 ± √4) / 2 y = (4 ± 2) / 2

This gives us two possible solutions:

y₁ = (4 + 2) / 2 = 3 y₂ = (4 - 2) / 2 = 1

Both solutions are already expressed to the nearest hundredth.

Solving for 'y' in Systems of Equations

Systems of equations involve multiple equations with multiple variables. To solve for 'y', you'll need to use methods like substitution or elimination.

Example 4: Solve for 'y' in the following system of equations:

x + y = 5 x - y = 1

Using the elimination method:

Add the two equations together: 2x = 6 => x = 3

Substitute x = 3 into either equation (let's use the first one):

3 + y = 5 => y = 2

Therefore, y = 2.00 to the nearest hundredth.

Solving for 'y' using Trigonometric Functions

Trigonometric functions relate the angles of a right-angled triangle to the ratios of its sides. Solving for 'y' might involve using functions like sine, cosine, or tangent.

Example 5: In a right-angled triangle, the hypotenuse is 10 units and the angle opposite side 'y' is 30 degrees. Find the length of side 'y'.

We use the sine function:

sin(30°) = y / 10 y = 10 * sin(30°) = 10 * 0.5 = 5

Therefore, y = 5.00.

Solving for 'y' in Exponential and Logarithmic Equations

Exponential equations involve variables in the exponent, while logarithmic equations are the inverse of exponential equations. Solving for 'y' often requires using logarithmic properties or exponential rules.

Example 6: Solve for y in the equation 2ʸ = 16

We can rewrite 16 as 2⁴:

2ʸ = 2⁴

Therefore, y = 4.00

Example 7: Solve for y in the equation log₂(y) = 3

This means 2 raised to the power of 3 equals y:

y = 2³ = 8

Therefore, y = 8.00

Solving for 'y' using Calculus

Calculus introduces techniques like differentiation and integration which can be used to find the value of 'y' in more complex scenarios, often involving rates of change or areas under curves. These methods are beyond the scope of a simple guide, but it's important to acknowledge their existence and relevance for advanced problems.

For instance, finding the maximum or minimum value of a function often involves finding the derivative and setting it to zero. The resulting value of 'x' can then be substituted back into the original function to find the corresponding value of 'y'.

Approximation Techniques

Sometimes, an exact solution for 'y' might be difficult or impossible to find. In such cases, approximation techniques can be employed to find a value of 'y' to the nearest hundredth. These techniques often involve iterative methods, such as the Newton-Raphson method, which repeatedly refines an initial guess to get closer to the actual solution.

Importance of Rounding to the Nearest Hundredth

Rounding to the nearest hundredth (two decimal places) is essential for several reasons:

• Precision: It provides a balance between accuracy and simplicity. While more decimal places offer greater precision, they are often unnecessary and can clutter the results.
• Standardization: Rounding to a specific number of decimal places helps to standardize results, making it easier to compare and interpret data from different sources.
• Practical Applications: Many real-world applications, like engineering and finance, require results to be rounded to a specific number of decimal places for practical reasons.

This comprehensive guide has covered various approaches to finding 'y' to the nearest hundredth. Remember to choose the appropriate method depending on the type of equation or function involved. Practice is key to mastering these techniques and developing the skills necessary for accurate and efficient problem-solving. By understanding the underlying principles and applying these strategies, you'll be able to confidently solve a wide range of mathematical problems. Remember to always double-check your work and consider using calculators or software for complex calculations to minimize errors and increase efficiency.