Understanding the Equation of a Line with a Slope of 1/4 Passing Through Point (4, 1)

When dealing with a line with a slope of 1/4 passing through the point (4, 1), finding the equation of the line in slope-intercept form (y mx b) involves several steps. Let's explore the process in detail.

Slope-Intercept Form: y mx b

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

y mx b

Step 1: Identify the Slope (m)

The slope of the line is given as 1/4. This means that for every 4 units of horizontal movement, the line rises 1 unit vertically.

Step 2: Use the Point (4, 1)

The line passes through the point (4, 1), which are the x and y coordinates, respectively. We can use this point to locate the line on the coordinate plane.

Step 3: Apply the Point-Slope Form

Alternative to the slope-intercept form, the point-slope form of a line is:

y - y_1 m(x - x_1)

Substitute the known values:

M 1/4, x_1 4, y_1 1

The equation becomes:

y - 1 (1/4)(x - 4)

Step 4: Simplify the Equation

First, distribute the slope:

y - 1 (1/4)x - (1/4)(4)

Now, simplify the equation:

y - 1 (1/4)x - 1

Next, solve for y:

y (1/4)x - 1 1

The final equation in slope-intercept form is:

y (1/4)x

Alternative Solution

If the slope (1/4) is inputted into the traditional line equation form:

y (1/4)x b

To find the y-intercept (b), substitute the known point (4, 1) into this equation:

1 (1/4)4 b

Calculate the value of b:

1 1 b

b 0

Therefore, the equation of the line is:

y (1/4)x 0

Which simplifies to:

y (1/4)x

Significance of Different Slopes (1.4 vs. 1/4)

The slope of the line, 1/4, is quite different from 1.4. Slope 1.4 means a steeper rise for every unit of run compared to 1/4. Understanding the precise slope is crucial for determining the correct equation of the line.

Now you can confidently use the slope-intercept (y mx b) and point-slope forms (y - y_1 m(x - x_1)) to determine the equation of a line given a specific slope and a point through which it passes.