Which Is The Graph Of 2x 4y 6

Which is the Graph of 2x + 4y = 6? A Comprehensive Guide

Understanding linear equations and their graphical representations is fundamental to algebra and numerous applications in various fields. This article delves into the equation 2x + 4y = 6, exploring its characteristics, how to graph it, and its real-world significance. We'll cover different methods for graphing this equation, interpreting its slope and intercepts, and connecting it to broader mathematical concepts.

Understanding the Equation 2x + 4y = 6

The equation 2x + 4y = 6 is a linear equation in two variables, x and y. A linear equation always represents a straight line when graphed. The equation is in standard form, Ax + By = C, where A, B, and C are constants. In this case, A = 2, B = 4, and C = 6.

This specific equation describes a relationship between x and y where, for any given value of x, there's a corresponding value of y that satisfies the equation. The graph visually represents all these paired values (x, y) that make the equation true.

Method 1: Finding the Intercepts

One of the simplest ways to graph a linear equation is by finding its x-intercept and y-intercept.

• x-intercept: The point where the line crosses the x-axis (where y = 0). To find it, set y = 0 in the equation and solve for x:

  2x + 4(0) = 6 2x = 6 x = 3

  The x-intercept is (3, 0).

• y-intercept: The point where the line crosses the y-axis (where x = 0). To find it, set x = 0 in the equation and solve for y:

  2(0) + 4y = 6 4y = 6 y = 6/4 = 3/2 = 1.5

  The y-intercept is (0, 1.5).

Now, plot these two points (3, 0) and (0, 1.5) on a coordinate plane. Draw a straight line passing through both points. This line represents the graph of the equation 2x + 4y = 6.

Method 2: Using the Slope-Intercept Form

Another effective method involves converting the equation into slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Let's rearrange the equation:

2x + 4y = 6 4y = -2x + 6 y = (-2/4)x + 6/4 y = (-1/2)x + 3/2

From this form, we can directly identify:

• Slope (m) = -1/2: This indicates that for every 2 units increase in x, y decreases by 1 unit. The negative slope signifies a downward-sloping line.
• y-intercept (b) = 3/2 = 1.5: This confirms our earlier calculation.

Starting from the y-intercept (0, 1.5), use the slope to find another point. Since the slope is -1/2, move 2 units to the right and 1 unit down, which brings you to the point (2, 1). Plot these two points and draw a line connecting them. This will be identical to the graph obtained using the intercept method.

Method 3: Using a Table of Values

Creating a table of values is a more systematic approach, especially useful for equations that might not easily yield intercepts. Choose a few values for x, substitute them into the equation, and solve for the corresponding y values.

Plot these points on a coordinate plane, and draw a line through them. Again, you'll obtain the same line as before.

Interpreting the Graph

The graph of 2x + 4y = 6 is a straight line with a negative slope, intersecting the x-axis at (3, 0) and the y-axis at (0, 1.5). This visual representation illustrates the infinite number of (x, y) pairs that satisfy the equation. Each point on the line represents a solution to the equation.

Real-World Applications

Linear equations like 2x + 4y = 6 have numerous real-world applications. For example:

• Economics: It could model the relationship between the quantity of two goods consumed (x and y) given a fixed budget (6).
• Physics: It could represent a linear relationship between two physical quantities.
• Engineering: It might describe a linear relationship between input and output variables in a system.
• Business: It could model simple cost functions or revenue projections.

Advanced Concepts and Extensions

• Systems of Equations: The equation 2x + 4y = 6 can be part of a system of linear equations. Solving such systems involves finding the point(s) where the lines intersect.
• Inequalities: Instead of an equation, we could have an inequality like 2x + 4y > 6 or 2x + 4y ≤ 6. Graphing these inequalities involves shading a region of the coordinate plane instead of just drawing a line.
• Linear Programming: In optimization problems, linear equations are often used to define constraints, and the goal is to find the optimal solution within these constraints.

Conclusion

Graphing the equation 2x + 4y = 6 provides a clear visual representation of the relationship between the variables x and y. Using different methods, such as finding intercepts, using the slope-intercept form, or creating a table of values, leads to the same graphical result: a straight line with a negative slope, crossing the axes at specific points. Understanding this simple linear equation forms a foundation for exploring more complex mathematical concepts and their applications in diverse fields. The ability to graphically represent and interpret linear equations is a crucial skill for success in numerous academic and professional settings. Mastering these techniques opens the door to a deeper understanding of linear algebra and its far-reaching implications. Remember to practice graphing various linear equations to solidify your understanding and build confidence in tackling more challenging problems.