Equation of a Line with Given Gradient and Point Passing Through It

In this guide, we will explore how to find the equation of a line when given its gradient (slope) and a point through which it passes. We will use various forms of the equation of a line to walk through the process step by step. This will be useful for students, educators, and professionals who need to understand and apply the equation of a line in different contexts.

Introduction to Line Equation Forms

The equation of a straight line can be expressed in several forms, including the point-slope form, slope-intercept form, and general form. The common forms include:

Point-Slope Form: ( y - y_1 m(x - x_1) ) Slope-Intercept Form: ( y mx b ) General Form: ( Ax By C 0 )

Problem: Given a Line with Gradient -4 and Passing Through the Point (-2, -3)

We are given a line with a gradient of -4 and a point that it passes through, (-2, -3). Let's find its equation using both the point-slope form and the slope-intercept form.

Point-Slope Form Approach

Identify the given values: Slope (Gradient), ( m -4 ) Point ( (x_1, y_1) (-2, -3) ) Substitute the values into the point-slope form:

y - (y1)  m(x - x1)y - (-3)  -4(x - (-2))y   3  -4(x   2)

Simplify and distribute the slope (-4):

y   3  -4x - 8y  -4x - 8 - 3y  -4x - 11

The equation of the line in point-slope form is ( y -4x - 11 ).

Slope-Intercept Form Approach

Start with the slope-intercept form:

y  mx   by  -4x   b

Substitute the point (-2, -3) into the equation:

-3  -4(-2)   b-3  8   bb  -3 - 8b  -11

Substitute the value of ( b ) back into the slope-intercept form:

y  -4x - 11

The equation of the line in slope-intercept form is ( y -4x - 11 ).

Alternative Approaches

Let's explore a few more approaches to finding the equation of the same line.

Using the Slope-Intercept Form with a Different Point

Assume the equation of the line is ( y mx c ):

-3  -4(-2)   c-3  8   cc  -3 - 8c  -11

Substitute ( c ) back into the slope-intercept form:

y  -4x - 11

The equation of the line remains ( y -4x - 11 ).

Using the General Form of the Line

The general form of a line is ( Ax By C 0 ). We can convert the slope-intercept form to the general form.

Starting with ( y -4x - 11 ): Move all terms to one side: ( y 4x 11 0 ) The general form of the line is ( 4x y 11 0 ).

Both forms of the equation are equivalent.

Understanding the Slope and Y-Intercept

In the equation ( y -4x - 11 ), the coefficient of ( x ) is the slope of the line, which is -4. This indicates that for every unit increase in ( x ), ( y ) decreases by 4 units. The constant term -11 is the y-intercept, which means the line crosses the y-axis at the point (0, -11).

Conclusion

Through various approaches, we have derived the equation of a line with a gradient of -4 and passing through the point (-2, -3). The equation is ( y -4x - 11 ). Understanding these forms and techniques will help in solving similar problems involving the equation of a line.