Question

How do you write the equation \[y = \dfrac{2}{3}x - 7 \] in standard form?

Answer 1

Hint: We will try to move the constant terms in one side and the constant terms on the other side of the equation, and then we will compare it with the standard form of an equation. On doing some calculation we get the required answer.

Formula used: Any linear equation can be written as two following way:
 \[1. \] Slope-intercept form of a linear equation
 \[2. \] Standard form of a linear equation
Most of the time, any linear equation is written in slope-intercept form.
 \[1. \] the general equation of slope-intercept form is as following:
 \[y = m.x + c \] , where \[y,x \] are variables of the equation and \[c \] is the constant term.
And, \[m \] is called the slope of the line.
 \[m \] can have positive value or negative value, or it can be any fractional value or any real number.
So, the slope tells the character of any straight line.
 \[2. \] The general form of standard form is as following:
 \[Ax + By = c \] .
But the above form shall follow the following points:
 \[I. \] ‘ \[A \] ’ must be a positive number, it cannot be a negative number.
 \[II. \] \[A,B \] and \[C \] must be integers, they cannot be fractional numbers.

Complete step-by-step solution:
It is given in the question that: \[y = \dfrac{2}{3}x - 7 \]
Now, multiply both the sides by \[3 \] , we get:
 \[ \Rightarrow 3y = 2x - 21 \] .
Now, taking all the variables to the L.H.S, we get:
 \[ \Rightarrow 3y - 2x = - 21 \]
After arrangements, we get:
 \[ \Rightarrow - 2x + 3y = - 21 \]
So, by comparing the above standard form of equation, we can say that the term \[A = - 2 \] , which is positive, and \[B = 3 \] , \[C = - 21 \] , which are integers too.

\[\therefore \] The standard form of the given equation is \[ - 2x + 3y = - 21. \]

Note: Points to remember:
We need to simplify any linear equation to get the variables and constant terms on the different sides of the equation.
Slope of any linear equation tells us the character of the equation and the standard form of an equation helps us the number of integral solutions that exist for the equation.