How to Write the Equation of a Line in Standard Form with Given Slope and Y-Intercept

When dealing with the equation of a line, one common task is transforming the equation from slope-intercept form into standard form. This is particularly useful for various applications in mathematics and real-world scenarios. This article will guide you through the process, ensuring the equation is in the standard form (Ax By C), where (A), (B), and (C) are integers, and (A) is positive.

Step-by-Step Guide

To write the equation of a line in standard form when given the slope and y-intercept, follow these steps:

1. Start with the Slope-Intercept Form

The slope-intercept form of a line is given by:

[begin{equation} y mx b end{equation}

where (m) is the slope and (b) is the y-intercept. For this example, the slope (m -frac{9}{2}) and the y-intercept (b frac{3}{5}).

2. Substitute the Given Values

Substitute the given values into the slope-intercept form:

[begin{equation} y -frac{9}{2}x frac{3}{5} end{equation}

3. Clear the Fractions

To eliminate the fractions, find a common denominator for the terms in the equation. In this case, the denominators are 2 and 5. The least common denominator (LCD) is 10. Multiply every term in the equation by 10:

[begin{equation} 10y 10left(-frac{9}{2}xright) 10left(frac{3}{5}right) end{equation}

This simplifies to:

[begin{equation} 10y -45x 6 end{equation}

4. Rearrange to Standard Form

Next, rearrange the equation to match the standard form (Ax By C), where (A), (B), and (C) are integers:

[begin{equation} 45x 10y 6 end{equation}

Since (A 45) is already positive, the equation is now in standard form.

Example with Given Values

Given the y-intercept is (frac{3}{5}) and the slope is (-frac{9}{2}), follow these steps:

Step 1: Start with the Slope-Intercept Form

Substitute the given values into the slope-intercept form:

[begin{equation} y -frac{9}{2}x frac{3}{5} end{equation}

Step 2: Clear the Fractions

Multiply every term by the least common denominator, which is 10:

[begin{equation} 10y -45x 6 end{equation}

Step 3: Rearrange to Standard Form

Rearrange the equation to match the standard form:

[begin{equation} 45x 10y 6 end{equation}

Variation in Standard Form Across Countries

The standard form of a line can sometimes vary depending on the region. In the USA, the standard form is (Ax By C), but in some other countries, such as India, the equation may be written in a different form:

[begin{equation} frac{6}{5} 2y - 9x end{equation}

This equation is still the same line, just in a different format. In India, the standard form might be (9x - 2y frac{6}{5} 0). The exact form can vary, but the underlying line represented by the equation remains the same.

Conclusion

Writing an equation of a line in standard form with a given slope and y-intercept involves a few simple steps. By following the guide provided above, you can easily transform the equation into the standard form. Remember, the key is to clear the fractions and rearrange the equation accordingly. Understanding these concepts is crucial for solving various problems involving linear equations.

Further Reading

For more detailed information and additional examples, you can explore further resources on the topic of linear equations and their forms. This guide should serve as a solid foundation for understanding and applying the standard form of a line equation.