3x 2y 10 In Slope Intercept Form

Transforming 3x + 2y = 10 into Slope-Intercept Form: A Comprehensive Guide

The equation 3x + 2y = 10 represents a straight line. While useful in its current form, converting it to slope-intercept form (y = mx + b) offers significant advantages for understanding and visualizing the line's characteristics. This form explicitly reveals the slope (m) and the y-intercept (b), crucial elements for graphing, analyzing, and comparing lines. This article will comprehensively guide you through the transformation process, explaining the underlying concepts and providing examples for a complete understanding.

Understanding Slope-Intercept Form (y = mx + b)

Before diving into the conversion, let's refresh our understanding of the slope-intercept form: y = mx + b.

• y: Represents the dependent variable, typically plotted on the vertical axis.
• x: Represents the independent variable, typically plotted on the horizontal axis.
• m: Represents the slope of the line. The slope 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. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line. The slope is calculated as the change in y divided by the change in x (rise over run).
• b: Represents the y-intercept, the point where the line crosses the y-axis (where x = 0).

Converting 3x + 2y = 10 to Slope-Intercept Form

The goal is to isolate 'y' on one side of the equation to match the y = mx + b format. Here's a step-by-step process:

1. Subtract 3x from both sides: This moves the 'x' term to the right side of the equation.

   3x + 2y - 3x = 10 - 3x

   This simplifies to:

   2y = -3x + 10

2. Divide both sides by 2: This isolates 'y' and gives us the slope-intercept form.

   2y / 2 = (-3x + 10) / 2

   This simplifies to:

   y = (-3/2)x + 5

Now we have our equation in slope-intercept form. Let's analyze the results:

• Slope (m) = -3/2: This indicates a negative slope, meaning the line descends from left to right. The slope of -3/2 means that for every 2 units increase in x, y decreases by 3 units.

• Y-intercept (b) = 5: This means the line crosses the y-axis at the point (0, 5).

Visualizing the Line: Graphing the Equation

With the slope-intercept form, graphing the line becomes straightforward. We already know two key pieces of information:

1. The y-intercept (5): Plot a point at (0, 5) on the y-axis.
2. The slope (-3/2): Starting from the y-intercept, use the slope to find another point. Since the slope is -3/2, move down 3 units (because it's negative) and to the right 2 units. This gives you a second point at (2, 2). You can also move up 3 units and to the left 2 units to get another point (-2, 8).

Connect these points with a straight line, and you've successfully graphed the equation 3x + 2y = 10.

Applications and Further Exploration

Understanding the slope-intercept form opens doors to various applications:

1. Finding Intercepts

We already found the y-intercept. To find the x-intercept (where the line crosses the x-axis, where y = 0), substitute y = 0 into the original equation or the slope-intercept form:

• Using the original equation: 3x + 2(0) = 10 => 3x = 10 => x = 10/3

• Using the slope-intercept form: 0 = (-3/2)x + 5 => (3/2)x = 5 => x = 10/3

The x-intercept is (10/3, 0).

2. Comparing Lines

The slope-intercept form makes it easy to compare the steepness and position of different lines. For instance, compare y = (-3/2)x + 5 to y = 2x + 1. The first line has a steeper negative slope and a higher y-intercept.

3. Parallel and Perpendicular Lines

Parallel lines have the same slope but different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other (e.g., if one line has a slope of m, a perpendicular line will have a slope of -1/m).

4. Real-World Applications

The slope-intercept form finds application in various real-world scenarios:

• Economics: Modeling supply and demand curves.
• Physics: Representing velocity and acceleration.
• Engineering: Designing slopes and gradients.
• Finance: Analyzing investment growth.

Solving Problems Involving the Equation

Let's explore some problems that demonstrate the utility of converting to slope-intercept form:

Problem 1: Find the point on the line where x = 4.

Substitute x = 4 into the slope-intercept form:

y = (-3/2)(4) + 5 = -6 + 5 = -1

Therefore, the point is (4, -1).

Problem 2: Determine if the point (6, -4) lies on the line.

Substitute x = 6 and y = -4 into the original equation (or the slope-intercept form):

3(6) + 2(-4) = 18 - 8 = 10

Since the equation holds true, the point (6, -4) lies on the line.

Problem 3: Find the equation of a line parallel to y = (-3/2)x + 5 and passing through the point (2, 3).

Parallel lines have the same slope. The slope is -3/2. Use the point-slope form (y - y1 = m(x - x1)) with the given point (2, 3):

y - 3 = (-3/2)(x - 2)

y - 3 = (-3/2)x + 3

y = (-3/2)x + 6

Conclusion

Transforming the equation 3x + 2y = 10 into slope-intercept form (y = (-3/2)x + 5) provides a powerful tool for understanding and working with linear equations. This form reveals the slope and y-intercept, simplifying graphing, analyzing, and solving problems related to the line. The ability to easily visualize and interpret the line's characteristics makes the slope-intercept form invaluable in various mathematical and real-world applications. This thorough guide equips you with the knowledge and skills to confidently tackle problems involving linear equations and their graphical representations. Remember, consistent practice is key to mastering this concept and applying it effectively in different contexts.