How to Find the Y-Intercept When Given a Point and the Slope

The y-intercept of a linear equation provides a key point where the line crosses the y-axis. If you have a point and the slope (gradient) of a line, you can determine the y-intercept using the point-slope form of the equation of a line. This guide will walk you through the step-by-step process to find the y-intercept.

Understanding the Point-Slope Form

The point-slope form of the equation of a line is:

y - y1 m(x - x1)

Where:

m is the slope, or gradient, of the line.(x1, y1) is a point on the line.

This form is particularly useful when you know a specific point and the slope of the line.

Finding the Y-Intercept

The y-intercept is the point where the line crosses the y-axis, which corresponds to where x 0. To find the y-intercept, follow these steps:

Step-by-Step Process

Identify the point and the slope: Let’s say the point is (x1, y1) and the slope is m. Use the point-slope form: Write the equation of the line in point-slope form using the identified point and slope. Rearrange to find the y-intercept: To find the y-intercept, substitute x 0 into the equation and solve for y. Determine the y-intercept: Simplify the equation to find the value of y when x 0.

Example 1

Suppose you have the point (2, 3) and the gradient m 4.

Apply the Steps

Identify the point and gradient: x1 2 y1 3 m 4 Use the point-slope form:

y - 3 4(x - 2)

Rearrange to find the y-intercept:

Rearrange to solve for y:

y 4x - 8 3

y 4x - 5

Determine the y-intercept:

The y-intercept, b, is the value of y when x 0:

b -5

Therefore, the y-intercept is -5.

Example 2

Suppose you have a point on the line (4, 5) and the slope is 3/4.

Apply the Steps

Identify the point and gradient: x1 4 y1 5 m 3/4 Use the point-slope form:

y - 5 3/4(x - 4)

Rearrange to find the y-intercept:

Rearrange to solve for y:

y (3/4)x - 3 5

y (3/4)x 2

Determine the y-intercept:

The y-intercept, c, is the value of y when x 0:

c 2

Therefore, the y-intercept is 2.

Conclusion

Determining the y-intercept is a crucial step in understanding the linear function of a line. By using the point-slope form and substituting x 0, you can find the y-intercept easily. This method is widely used in mathematics and real-world applications such as economics, physics, and engineering.

Additional Resources

For additional practice and further understanding, refer to:

Mathplanet: Finding the Slope and the Y-Intercept Khan Academy: Finding the Y-Intercept from an Equation or a Chart