Solve 3x 4y 12 For Y

Solving for 'y': A Comprehensive Guide to 3x + 4y = 12

This article provides a comprehensive guide on how to solve the equation 3x + 4y = 12 for 'y'. We'll explore various methods, delve into the underlying algebraic principles, and even examine the graphical representation of this linear equation. Understanding how to manipulate equations like this is fundamental in algebra and has wide-ranging applications in various fields.

Understanding the Equation: 3x + 4y = 12

The equation 3x + 4y = 12 represents a linear equation in two variables, x and y. A linear equation is characterized by its straight-line graph when plotted on a Cartesian coordinate system. This equation signifies a relationship where the sum of three times 'x' and four times 'y' always equals 12. Our goal is to isolate 'y', expressing it in terms of 'x'. This will give us a formula that allows us to calculate the value of 'y' for any given value of 'x'.

Method 1: Isolating 'y' through Algebraic Manipulation

This is the most common and straightforward method. We'll use basic algebraic operations to isolate 'y' on one side of the equation.

Step 1: Subtract 3x from both sides:

The objective is to move the term containing 'x' to the right-hand side of the equation. To do this, we subtract 3x from both sides, maintaining the equation's balance:

3x + 4y - 3x = 12 - 3x

This simplifies to:

4y = 12 - 3x

Step 2: Divide both sides by 4:

Now, we need to isolate 'y' by eliminating the coefficient 4. We achieve this by dividing both sides of the equation by 4:

4y / 4 = (12 - 3x) / 4

This simplifies to:

y = (12 - 3x) / 4

Step 3: Simplify (Optional):

While the above equation is perfectly valid, we can further simplify it by distributing the division:

y = 12/4 - (3x)/4

This simplifies to:

y = 3 - (3/4)x or equivalently y = - (3/4)x + 3

This is the slope-intercept form of the equation (y = mx + c), where 'm' represents the slope (-3/4) and 'c' represents the y-intercept (3).

Method 2: Using the Addition and Subtraction Properties of Equality

This method highlights the fundamental properties of equality. Every step is justified by either the addition or subtraction property of equality, which states that adding or subtracting the same quantity from both sides of an equation preserves the equality.

Step 1: Apply the Subtraction Property of Equality:

We start with the original equation: 3x + 4y = 12. We want to isolate the term with 'y', so we subtract 3x from both sides:

3x + 4y - 3x = 12 - 3x

This results in:

4y = 12 - 3x

Step 2: Apply the Division Property of Equality:

Now, we need to isolate 'y'. We divide both sides by 4, using the division property of equality (dividing both sides by the same non-zero number maintains the equality):

4y / 4 = (12 - 3x) / 4

This leads us to the solution:

y = 3 - (3/4)x

Method 3: Graphical Representation and Interpretation

The equation 3x + 4y = 12 can be represented graphically as a straight line. The solution we derived, y = 3 - (3/4)x, provides valuable information about this line:

• Slope (m): The slope is -3/4. This indicates that for every 4-unit increase in x, y decreases by 3 units. The negative slope signifies a downward-sloping line.

• Y-intercept (c): The y-intercept is 3. This means the line intersects the y-axis at the point (0, 3).

To graph this line, you can plot the y-intercept (0, 3) and then use the slope to find another point. For example, if you increase x by 4 (moving to x=4), y will decrease by 3 (becoming y=0), giving you the point (4, 0). Draw a straight line through these two points, and you've graphically represented the solution. Any point on this line satisfies the original equation 3x + 4y = 12.

Applications and Further Exploration

Solving linear equations like 3x + 4y = 12 is a fundamental skill with diverse applications:

• Modeling Real-World Scenarios: Linear equations can model various real-world relationships. For example, this equation could represent the relationship between the number of hours worked (x) and the total earnings (y) at a certain hourly wage.

• System of Equations: Understanding how to solve for a variable in a linear equation is crucial when dealing with systems of equations, where multiple linear equations are solved simultaneously.

• Calculus and Beyond: The foundation of linear equations is essential for more advanced mathematical concepts like calculus and linear algebra.

Solving for x: A Complementary Perspective

While this article focused on solving for 'y', it's equally important to understand how to solve for 'x'. The process is similar, using inverse operations to isolate 'x':

Step 1: Subtract 4y from both sides:

3x + 4y - 4y = 12 - 4y

This simplifies to:

3x = 12 - 4y

Step 2: Divide both sides by 3:

3x / 3 = (12 - 4y) / 3

This simplifies to:

x = 4 - (4/3)y

This equation expresses 'x' in terms of 'y', providing another perspective on the same linear relationship.

Conclusion: Mastering Linear Equations

Solving the equation 3x + 4y = 12 for 'y' (or 'x') is a cornerstone of algebra. Mastering this process involves not only the mechanical application of algebraic operations but also a deeper understanding of the underlying principles of equality and the graphical interpretation of the solution. The ability to manipulate and solve linear equations is a fundamental skill with widespread applications in mathematics and various other fields. Practice and understanding the different methods will solidify your comprehension and enable you to confidently tackle more complex algebraic problems in the future. Remember to always check your solutions by substituting the values back into the original equation to verify their accuracy.