Understanding Slope-Intercept Form: Converting 2x-7y-21

When working with linear equations, it's often useful to represent them in slope-intercept form, which is y mx b. In this form, m represents the slope of the line, and b is the y-intercept. We will explore how to convert the equation 2x - 7y -21 into slope-intercept form and understand the significance of slope and y-intercepts.

Converting 2x - 7y -21 into Slope-Intercept Form

To convert the equation 2x - 7y -21 into slope-intercept form:

Isolate y on one side of the equation: Negate and move the terms involving x to the right side: Divide every term by -7 to solve for y.

Let's go through these steps in detail:

Step-by-Step Conversion

2x - 7y -21 Negate and move 2x to the right: Divide every term by -7:

The equation now in slope-intercept form:

Slope-Intercept Form

2x - 7y -21 converts to y 2/7x 3.

Now, we can identify the slope and the y-intercept from this equation.

Slope and Y-Intercept

Slope (m): The coefficient of x is 2/7. Therefore, m 2/7.

Y-Intercept (b): The constant term, when x is zero, is 3. Therefore, b 3.

Thus, the slope is 2/7 and the y-intercept is 3.

Alternative Method for Finding Slope and Intercept

There's a faster method to find the slope and intercepts of a linear equation when given in standard form Ax By C.

Slope: m -A/B. For our equation 2x - 7y -21, A 2 and B -7. Therefore, m -2/(-7) 2/7. Y-Intercept: b C/B. For our equation, C -21 and B -7. Therefore, b -21/(-7) 3.

Thus, the slope is 2/7, and the y-intercept is 3.

Conclusion

By converting the equation 2x - 7y -21 into slope-intercept form, we have obtained the equation in the form y 2/7x 3. This shows that the slope of the line is 2/7 and the y-intercept is 3. Both methods provide the same results, offering a clear and systematic approach to understanding linear equations.

Key Takeaways

The slope-intercept form is y mx b. To find the slope and y-intercept, isolate y in the standard form equation. Alternatively, use the formulas m -A/B and b C/B.

By mastering these concepts, you can easily manipulate linear equations and understand their graphical representations.