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Straight Lines: General equation of a line

Linear equations contain two variables and when we plot all the (x,y) pairs that make the equation for a line.

The formula for Standard Form is:

Here, A, B and C are real numbers and A≠ and B≠0.

Since the slope of a vertical line is undefined we don’t write the equation of a vertical line using either the slope-intersect form or the point-slope form. But we can use the standard form to express vertical lines.

To find where the line crosses each axis, let the other value be zero.

  • If y is zero, we have Ax = C, C/A is the x-intercept.
  • If x is zero, we have By = C, C/B is the y-intercept.
  • − A/B is the slope.

Using the x-intercept and/or y-intercept as a starting point, you can use the slope to graph more points on the line of the equation.

Write an equation of the line in STANDARD/GENERAL FORM using the information given: m=2 and (3, -2)

Start with Point-Slope Form

Now put into slope-intercept form

Now put into the general form:

Leading negatives are not recommended. Therefore, change all the signs of each term

Write an equation of the line in STANDARD FORM using the information given: (-4, 4) and (0, 3)

Start with Point-Slope Form

Now put into slope-intercept form

Now put into Standard form

The process of simplifying any type of line equations to the general form is as follows:

  1. Do all the possible algebraic operations ( multiplication division addition and subtraction)
  2.  Find the determinant in case of  two-point form
  3.  evaluate all the trigonometric ratios in the case of normal form
  4.  eliminate the parameter in case of two intercept form and slope-intercept form
  5.  shift all the gums on one side of the equation
  6.  Arrange the terms by algebraic operations and present them in this style
Equation formEquationNotes
Slope-intercepty = mx + bm is the slope.

b is the y-intercept.

Point slopem is the slope and (x1, y1) is a point on the line
Normal formp is the perpendicular from the origin to the line and α is the angle between this perpendicular and the x-axis.
Intercept forma= x-intercept and b= y-intercept
Parallel to axisVertical: x=a and Horizontal y=ba= x-intercept and b= y-intercept
Two-point formThe line passes through A (x1, y1) and B (x2, y2)
Standard formA is positive. A, B and C are real numbers and A≠ and B≠0.

 

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