Rewrite The Following Equation In Slope Intercept Form

Rewriting Equations in Slope-Intercept Form: A Comprehensive Guide

The slope-intercept form of a linear equation is a fundamental concept in algebra, providing a clear and concise way to represent a straight line. Understanding this form is crucial for various mathematical applications, from graphing lines to solving systems of equations. This comprehensive guide will delve deep into rewriting equations into the slope-intercept form, exploring various methods and tackling diverse equation types.

What is Slope-Intercept Form?

The slope-intercept form of a linear equation is expressed as:

y = mx + b

Where:

• 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, indicating its steepness and direction. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend.
• b represents the y-intercept, indicating the point where the line crosses the y-axis (the value of y when x = 0).

This form is incredibly useful because it immediately tells us two key characteristics of the line: its slope and its y-intercept. This information allows for quick graphing and easy analysis of the line's properties.

Methods for Rewriting Equations into Slope-Intercept Form

Several methods can be used to rewrite equations into the slope-intercept form. The most common methods include:

1. Solving for y: The Fundamental Approach

This is the most straightforward method. It involves manipulating the equation algebraically until 'y' is isolated on one side of the equation, leaving the equation in the form y = mx + b.

Example 1: Rewrite the equation 2x + 3y = 6 into slope-intercept form.

1. Subtract 2x from both sides: 3y = -2x + 6
2. Divide both sides by 3: y = (-2/3)x + 2

Therefore, the slope-intercept form is y = (-2/3)x + 2. Here, the slope (m) is -2/3, and the y-intercept (b) is 2.

Example 2: Rewrite the equation x - 4y = 8 into slope-intercept form.

1. Subtract x from both sides: -4y = -x + 8
2. Divide both sides by -4: y = (1/4)x - 2

Therefore, the slope-intercept form is y = (1/4)x - 2. The slope is 1/4, and the y-intercept is -2.

2. Using Point-Slope Form as an Intermediate Step

The point-slope form of a linear equation is given by:

y - y₁ = m(x - x₁)

where (x₁, y₁) is a point on the line, and m is the slope. If you know the slope and a point on the line, you can use the point-slope form to derive the slope-intercept form.

Example 3: A line has a slope of 2 and passes through the point (1, 3). Rewrite its equation in slope-intercept form.

1. Substitute the values into the point-slope form: y - 3 = 2(x - 1)
2. Expand and simplify: y - 3 = 2x - 2
3. Add 3 to both sides: y = 2x + 1

The slope-intercept form is y = 2x + 1.

3. Using Two Points to Find the Slope and Y-Intercept

If you know two points that lie on the line, you can first calculate the slope using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Then, substitute the slope and one of the points into the point-slope form and simplify to obtain the slope-intercept form.

Example 4: Find the slope-intercept form of the line passing through points (2, 4) and (4, 8).

1. Calculate the slope: m = (8 - 4) / (4 - 2) = 4 / 2 = 2
2. Use the point-slope form with (2, 4): y - 4 = 2(x - 2)
3. Simplify: y - 4 = 2x - 4
4. Add 4 to both sides: y = 2x

The slope-intercept form is y = 2x. Note that the y-intercept is 0 in this case.

Handling Different Equation Types

Not all equations are initially presented in a readily convertible format. Let's examine how to handle some variations:

1. Equations with Fractions

Equations containing fractions can be simplified before solving for y. Multiplying both sides by the least common denominator (LCD) can eliminate the fractions.

Example 5: Rewrite the equation (1/2)x + (2/3)y = 1 into slope-intercept form.

1. Find the LCD (6): Multiply both sides by 6: 3x + 4y = 6
2. Solve for y: 4y = -3x + 6
3. Divide by 4: y = (-3/4)x + (3/2)

The slope-intercept form is y = (-3/4)x + (3/2).

2. Equations with Decimals

Similar to fractions, equations with decimals can be simplified by multiplying both sides by a power of 10 to eliminate the decimals.

Example 6: Rewrite the equation 0.5x + 0.2y = 1.5 into slope-intercept form.

1. Multiply both sides by 10: 5x + 2y = 15
2. Solve for y: 2y = -5x + 15
3. Divide by 2: y = (-5/2)x + (15/2)

The slope-intercept form is y = (-5/2)x + (15/2).

3. Vertical and Horizontal Lines

• Vertical Lines: Vertical lines have undefined slopes and are represented by equations of the form x = c, where 'c' is a constant. These cannot be written in slope-intercept form.
• Horizontal Lines: Horizontal lines have a slope of 0 and are represented by equations of the form y = c, where 'c' is a constant. This is already in slope-intercept form (m = 0, b = c).

Applications of Slope-Intercept Form

The slope-intercept form has numerous applications:

• Graphing Linear Equations: Knowing the slope and y-intercept allows for quick and accurate graphing of the line.
• Predicting Values: Given a value for x, one can easily predict the corresponding value of y.
• Comparing Linear Relationships: The slopes and y-intercepts of different lines can be compared to analyze their relative steepness and starting points.
• Solving Systems of Linear Equations: The slope-intercept form is often used in solving systems of equations graphically or algebraically.
• Real-world Modeling: Linear equations in slope-intercept form are used to model various real-world phenomena, such as the relationship between distance and time, cost and quantity, and many more.

Conclusion

Rewriting equations into slope-intercept form is a fundamental skill in algebra. Mastering the various techniques discussed in this guide will equip you with a powerful tool for analyzing and understanding linear relationships, solving problems efficiently, and applying mathematical concepts to real-world scenarios. Remember to practice regularly to reinforce your understanding and improve your speed and accuracy. The more you practice, the easier it will become to identify the quickest and most efficient approach for each equation. From simple equations to those involving fractions or decimals, consistent practice will build your confidence and make this a straightforward process.