Standard form of a linear equation

Linear equation

Linear equations allow us to plot the variables on a line graph when solved. The most common linear equations have two unsolved variables, usually represented by an X and Y. To write a linear equation in standard form, you must format it as ax + by = c.

• Assess your equation to figure out how to set the equation in standard form.
• Move the C term to the other side of the equation.
• Group the ax and by integers together on the same side.

Standard form

The Standard Form for a linear equation in two variables, x and y, is usually given as Ax + By = C

where, if at all possible, A, B, and C are integers, and A is nonnegative, and, A, B, and C have no common factors other than 1. If we have a linear equation in slopeintercept form,

y = m x + b

we can change that equation into Standard Form. To do this we need to express the slope and the ordinate of the the yintercept in rational number form, that is, as the quotient of two integers. For the kinds of problems that we usually find in math classes, this is not much of a demand. The slope is defined to be the change in y divided by the change in x.

We can transform slopeintercept form equations into standard form equations. Why we want to do this because standard form allows us to write the equations for vertical lines, which is not possible in slopeintercept form.

A second reason for putting equations into standard form is that it allows us to employ a technique for solving systems of linear equations.

A third reason to use standard form is that it simplifies finding parallel and perpendicular lines.

We will write the given equation in standard form as follows:

• If there are fractions:

  Multiply each term in the equation by the LCD. Add or subtract to get x and y on the same side and the number term on the opposite side.

• If there are decimals:

  Multiply each term in the equation by 10, 100, 1000, etc.Add or subtract to get x and y on the same side and the number term on the opposite side.

• If there are no fractions nor decimals:

  Add or subtract to get x and y on the same side and the number term on the opposite side.

  Slope-intercept equations can easily be changed to standard form. Consider the equation: y = 3x - 1 - 3x + y = - 1 Subtract -3x from each side, satisfying Ax + By = C 3x - y = 1 Multiply the entire equation by - 1, satisfying A > 0 A and B are already integers, so we don't have to worry about changing them. In the standard form of an equation, the slope is always equal to -A/B