Standard Form of a Linear Equation

The standard form, also known as the general form of a linear equation, is one of the three ways in which we can express linear equations. The other two are the slope-intercept form and the point-slope form. 

The standard form of linear equation in one variable is given by:

Ax + B = 0

The standard form of linear equation in two variables is given by:

Ax + By = C

Here,

• A, B, and C are the integers, where A is a positive integer
• x and y are variables

Here is an example of a linear equation in the standard form:

7x – 3y = 16

Here, A = 7, B = -3, and C = 16 (all are integers), and A > 0.

The standard form of a linear equation is used in solving systems of equations, where the equations are arranged one above the other for further simplification. It clearly shows the relationship between the variables.

These equations are also used for graphing by first converting to the slope-intercept form. 

Converting to Standard Form

The slope-intercept form is expressed as y = mx + b, and the point-slope form is expressed as (y – y₁) = m(x – x₁). 

From Slope-Intercept Form 

Let us convert the equation ${y=\dfrac{2}{3}x-4}$ to its standard form.

Step 1: Simplifying 

Multiplying both sides by 3 to eliminate the fraction,

⇒ 3y = 2x – 12

Step 2: Moving the Variable Terms to One Side 

If the linear equation is already in simplified form, we rewrite the terms to match the standard form of the linear equation.

⇒ 3y – 2x = -12

⇒ -2x + 3y = -12

Step 3: Rearranging the Terms

Since A is negative, we multiply the equation by -1 to make it positive.

⇒ 2x – 3y = 12

This is the standard form of the given equation.

From Point-Slope Form 

Let us convert the equation ${y-4=\dfrac{1}{2}\left( x-6\right)}$ to its corresponding standard form.

Step 1: Simplifying 

Multiplying both sides by 2 to eliminate the fraction,

⇒ 2(y – 4) = x – 6

⇒ 2y – 8 = x – 6

Step 2: Moving the Variable Terms to One Side 

Now, by making the left-hand side of the equation the same as the standard form,

⇒ 2y – x = -6 + 8

⇒ -x + 2y = 2

Step 3: Rearranging the Terms

Since A is negative, we multiply the equation by -1 to make it positive.

⇒ x – 2y = -2

This is the standard form of the given equation.

Graphing

To graph an equation from its standard form, we first need to find its x- and y-intercepts. These points show where the line crosses the x-axis and y-axis, respectively. 

Let us graph the equation: 4x + 3y = 12

Step 1: Finding the X-Intercept

Since at the x-intercept, y = 0

Now, substituting y = 0 into the equation, we get

4x + 3y = 12

⇒ 4x + 3(0) = 12

⇒ 4x = 12

Solving for x, we get

⇒ x = 3

Thus, the x-intercept is (3, 0)

Step 2: Finding the Y-Intercept 

Since at the y-intercept, x = 0

Now, substituting x = 0 into the equation, we get

4x + 3y = 12

⇒ 4(0) + 3y = 12

⇒ 3y = 12

Solving for y, we get

⇒ y = 4

Thus, the y-intercept is (0, 4)

Step 3: Plotting the Intercepts and Drawing the line 

When plotting the points (3, 0) and (0, 4) on the graph, we get

Here, the line represents the equation 4x + 3y = 12, and every point on it is a solution to the equation.

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