3x 2y 8 Slope Intercept Form

Decoding the 3x + 2y = 8 Equation: A Deep Dive into Slope-Intercept Form

The equation 3x + 2y = 8 represents a linear relationship between two variables, x and y. While presented in standard form, understanding its slope-intercept form (y = mx + b) unlocks deeper insights into its graphical representation and characteristics. This comprehensive guide will meticulously explore the transformation process, the significance of the slope and y-intercept, and practical applications of this linear equation.

From Standard Form to Slope-Intercept Form: The Transformation

The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. Our equation, 3x + 2y = 8, fits this mold perfectly. To convert it to slope-intercept form (y = mx + b, where 'm' is the slope and 'b' is the y-intercept), we need to isolate 'y'. Let's walk through the steps:

1. Subtract 3x from both sides: This isolates the term containing 'y'. The equation becomes: 2y = -3x + 8

2. Divide both sides by 2: This isolates 'y', giving us the slope-intercept form: y = (-3/2)x + 4

Now we have the equation in its slope-intercept form: y = (-3/2)x + 4. This simple transformation provides crucial information about the line's characteristics.

Understanding the Slope (m = -3/2)

The slope, 'm', represents the rate of change of y with respect to x. In simpler terms, it indicates the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means it falls from left to right. In our equation, the slope is -3/2.

• Negative Slope: The negative sign indicates that the line slopes downwards from left to right.

• Magnitude of the Slope: The fraction -3/2 signifies that for every 2 units increase in x, y decreases by 3 units. This provides a precise measure of the line's steepness. A larger absolute value of the slope indicates a steeper line.

Understanding the Y-intercept (b = 4)

The y-intercept, 'b', represents the point where the line intersects the y-axis (where x = 0). In our equation, the y-intercept is 4. This means the line crosses the y-axis at the point (0, 4).

Graphing the Equation: Visualizing the Line

With the slope and y-intercept in hand, graphing the equation becomes straightforward.

1. Plot the y-intercept: Begin by plotting the point (0, 4) on the coordinate plane.

2. Use the slope to find another point: Since the slope is -3/2, move 2 units to the right and 3 units down from the y-intercept. This gives us the point (2, 1).

3. Draw the line: Draw a straight line passing through the points (0, 4) and (2, 1). This line represents the equation 3x + 2y = 8.

Finding the X-intercept: Where the Line Crosses the X-Axis

The x-intercept is the point where the line crosses the x-axis (where y = 0). To find it, substitute y = 0 into the original equation or the slope-intercept form:

Using the original equation: 3x + 2(0) = 8 => 3x = 8 => x = 8/3

Using the slope-intercept form: 0 = (-3/2)x + 4 => (3/2)x = 4 => x = 8/3

Therefore, the x-intercept is (8/3, 0).

Applications of the Equation: Real-World Scenarios

Linear equations like 3x + 2y = 8 have numerous real-world applications. Consider these examples:

• Cost Modeling: Suppose x represents the number of hours worked and y represents the total cost. The equation could model a scenario where a fixed cost of $4 (y-intercept) is incurred, plus an hourly rate of $1.50 (|slope|).

• Mixing Solutions: The equation could represent the mixing of two solutions with different concentrations, where x and y represent the volumes of each solution.

• Resource Allocation: The equation could model the allocation of resources, with x and y representing the quantities of two different resources.

Extending the Understanding: Parallel and Perpendicular Lines

The slope plays a crucial role in determining the relationship between different lines.

• Parallel Lines: Parallel lines have the same slope. Any line parallel to y = (-3/2)x + 4 will also have a slope of -3/2.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -3/2 is 2/3. Any line perpendicular to y = (-3/2)x + 4 will have a slope of 2/3.

Solving Systems of Equations Involving 3x + 2y = 8

The equation 3x + 2y = 8 can be used in conjunction with other linear equations to solve systems of equations. This involves finding the point (x, y) that satisfies both equations simultaneously. Common methods for solving systems of equations include substitution, elimination, and graphing.

For instance, consider the system:

3x + 2y = 8 x - y = 1

We can solve this system using substitution or elimination to find the unique solution (x, y) where both equations intersect.

Analyzing the Equation's Domain and Range

The domain of a linear equation like 3x + 2y = 8 is all real numbers for x, as there's no restriction on the values x can take. Similarly, the range is all real numbers for y. This means the line extends infinitely in both the x and y directions.

Conclusion: Mastering the 3x + 2y = 8 Equation

The seemingly simple equation 3x + 2y = 8 offers a rich understanding of linear equations. By transforming it into slope-intercept form, we uncover essential information about its slope, y-intercept, and graphical representation. This knowledge extends to applications in various fields and provides a solid foundation for tackling more complex mathematical concepts involving linear relationships. Understanding the intricacies of this equation empowers you to analyze, interpret, and apply linear relationships in diverse real-world scenarios. The ability to swiftly convert between standard form and slope-intercept form is a valuable skill in algebra and beyond.