Equation of a Straight Line with a Gradient of -4 and a Y-Intercept of -3

When working with linear equations, the slope-intercept form is a particularly useful representation. This form is given by Y mx b, where m is the gradient (slope) and b is the y-intercept (the value of Y when X is 0).

The Equation in Slope-Intercept Form

Given a line with a slope m -4 and a y-intercept c -3, we can write the equation of the line in slope-intercept form as:

Y -4X - 3

Deriving the Equation from Given Points

To derive the equation, we can use the slope-intercept form Y - Y1 m(X - X1). Given that the y-intercept is -3, the point on the line when X 0 is (0, -3).

Let's use this information to find the equation:

Y - (-3) -4(X - 0)
Y 3 -4X
Y -4X - 3

This confirms the equation in slope-intercept form is Y -4X - 3.

Understanding the Graphical Representation

The graph of the line will have the y-intercept at (0, -3) and a slope of -4. This means for every 1 unit increase in X, Y decreases by 4 units.

Using the form Y mx b, we can verify the equation again. Given m -4 and Y intercept -3, the equation remains:

Y -4X - 3.

Another way to express this is:

Y -4X - 3. You can also write -4 over 1 as -4/1 if needed.

Forms of Linear Equations

There are several forms to describe a line:

tThe standard form is AX BY C 0. tThe point-slope form is Y - Y1 m(X - X1), used when given a point (X1, Y1) and a slope m. tThe slope-intercept form is Y mx b, used when given a slope m and a y-intercept b.

For the given problem, the equation in slope-intercept form is:

Y -4X - 3.