12x 4y 20 Solve For Y

Solving for 'y': A Comprehensive Guide to 12x + 4y = 20

The equation 12x + 4y = 20 represents a linear relationship between two variables, x and y. Solving for 'y' means isolating 'y' on one side of the equation, expressing it in terms of 'x'. This process is fundamental in algebra and has numerous applications in various fields, from physics and engineering to economics and finance. This guide will walk you through solving this specific equation and then explore broader concepts related to solving for variables in linear equations.

Understanding the Equation

Before diving into the solution, let's understand what the equation 12x + 4y = 20 signifies. This equation represents a straight line on a graph. Each point (x, y) that satisfies this equation lies on that line. Solving for 'y' allows us to express the relationship explicitly – giving us the y-coordinate for any given x-coordinate.

Step-by-Step Solution: Isolating 'y'

The goal is to manipulate the equation to get 'y' by itself on one side of the equals sign. We'll achieve this using basic algebraic operations.

1. Subtract 12x from both sides:

   This eliminates '12x' from the left side, leaving only the term with 'y'.

   12x + 4y - 12x = 20 - 12x

   This simplifies to:

   4y = 20 - 12x

2. Divide both sides by 4:

   This isolates 'y' by removing the coefficient '4'.

   4y / 4 = (20 - 12x) / 4

   This simplifies to:

   y = 5 - 3x

Therefore, the solution for 'y' in the equation 12x + 4y = 20 is y = 5 - 3x.

This equation is now in slope-intercept form (y = mx + b), where:

• m represents the slope of the line (-3 in this case). The slope indicates the steepness and direction of the line. A negative slope means the line is decreasing from left to right.
• b represents the y-intercept (5 in this case). The y-intercept is the point where the line crosses the y-axis.

Visualizing the Solution: Graphing the Line

Plotting this equation on a graph provides a visual representation of the relationship between x and y. The line will have a slope of -3 and intersect the y-axis at 5.

You can choose several values for 'x', substitute them into the equation y = 5 - 3x, and calculate the corresponding 'y' values to plot the points on the graph. Connecting these points will reveal the straight line representing the equation.

For example:

• If x = 0, y = 5 - 3(0) = 5 (Point: (0, 5))
• If x = 1, y = 5 - 3(1) = 2 (Point: (1, 2))
• If x = 2, y = 5 - 3(2) = -1 (Point: (2, -1))
• If x = -1, y = 5 - 3(-1) = 8 (Point: (-1, 8))

By plotting these points and drawing a line through them, you can visually confirm the solution and understand the linear relationship depicted by the equation.

Applications of Solving Linear Equations

Solving for a variable in a linear equation, like the one we've solved, is crucial across many disciplines:

• Physics: Analyzing motion, forces, and other physical phenomena often involves solving linear equations. For example, determining the velocity of an object given its acceleration and time requires solving a linear equation.

• Engineering: Designing structures, circuits, and systems relies heavily on linear equations. Engineers use these equations to model and predict the behavior of their designs.

• Economics: Economic models frequently utilize linear equations to represent relationships between variables like supply and demand, cost and revenue.

• Finance: Calculating interest, loan repayments, and investment returns involves solving linear equations.

• Computer Science: Linear equations are used in algorithms for graphics, simulations, and machine learning.

Solving for Other Variables

While we solved for 'y' in this specific case, it's equally important to understand how to solve for 'x'. Let's rearrange the original equation, 12x + 4y = 20, to solve for 'x':

1. Subtract 4y from both sides:

   12x + 4y - 4y = 20 - 4y

   12x = 20 - 4y

2. Divide both sides by 12:

   12x / 12 = (20 - 4y) / 12

   x = (20 - 4y) / 12

   This can be simplified further by dividing each term by 4:

   x = (5 - y) / 3

Now we have an expression for 'x' in terms of 'y'. This shows the flexibility of linear equations; we can solve for any variable given the others.

Handling More Complex Linear Equations

The principles used to solve 12x + 4y = 20 extend to more complex linear equations. These might involve:

• Fractions: Equations with fractions require finding a common denominator before solving.

• Decimals: Similar to fractions, decimals can be handled by multiplying the equation by a suitable power of 10 to eliminate the decimal points.

• Parentheses: Equations with parentheses require simplifying the expression within the parentheses first using the distributive property (e.g., a(b + c) = ab + ac).

Regardless of the complexity, the underlying strategy remains the same: use algebraic operations (addition, subtraction, multiplication, division) to isolate the variable you want to solve for.

Practical Applications and Real-World Scenarios

Let's imagine a scenario where this equation could be applied:

A bakery sells two types of cookies: chocolate chip (x) and oatmeal raisin (y). Chocolate chip cookies cost $12 per dozen, and oatmeal raisin cookies cost $4 per dozen. If a customer spends $20 on a combination of both types of cookies, the equation 12x + 4y = 20 represents this scenario.

Solving for 'y' (y = 5 - 3x) allows the bakery to determine how many dozens of oatmeal raisin cookies a customer can buy, depending on the number of chocolate chip cookies purchased. For instance, if the customer buys one dozen chocolate chip cookies (x = 1), they can buy two dozen oatmeal raisin cookies (y = 2).

Conclusion: Mastering Linear Equations

Solving for 'y' in the equation 12x + 4y = 20, resulting in y = 5 - 3x, is a fundamental algebraic skill with vast practical implications. Understanding this process lays a strong foundation for tackling more complex mathematical problems across various fields. Mastering the techniques discussed here will equip you to confidently solve linear equations and interpret their meaning within real-world contexts. Remember to always practice and apply these techniques to further enhance your understanding and problem-solving abilities.