Question

How do you write an equation in standard form given point $\left( {0,0} \right)$ and slope $\dfrac{1}{3}$.

Answer 1

As the point and the slope of the line is given. First, substitute the point and slope in the slope-intercept formula, $y = mx + c$ to find the y-intercept. After that use the slope and y-intercept to find the equation of the line. Then move all variables on one side and the constant on another side as $Ax + By = C$ to make the equation in standard form.

Complete step-by-step answer:
The given point is $\left( {0,0} \right)$ and the slope is $\dfrac{1}{3}$.
As we know that the equation of the line in slope-intercept form is,
$y = mx + c$
Where,
m is the slope of the line and c is the y-intercept.
Substituting the point $\left( {0,0} \right)$ and the slope $\dfrac{1}{3}$ in slope-intercept form, we get
$\Rightarrow 0 = \dfrac{1}{3} \times 0 + c$
On simplifying the terms, we get
$\Rightarrow c = 0$
Now, we have a slope $m = \dfrac{1}{3}$ and y-intercept $c = 0$.
Use these values to find the equation of the line.
Substitute these values in slope-intercept form,
$\Rightarrow y = \dfrac{1}{3}x + 0$
Simplify the terms,
$\Rightarrow y = \dfrac{1}{3}x$
Multiply both sides by 3 to remove the rational part,
$\Rightarrow 3y = x$
So, the equation of the line is $x = 3y$.
Now, we have to write the equation in standard form.
The standard equation or general form of linear equation with two variables is,
$Ax + By = C$
Where A, B, C are real numbers while x and y are variables. Here, A and B are not equal to zero.
Now, subtract $3y$ from both sides,
$\Rightarrow x - 3y = 3y - 3y$
On simplifying the terms, we get
$\therefore x - 3y = 0$

Hence, the equation in standard form is $x - 3y = 0$.

Note:
Linear equation represents a straight line. Linear equations in two variables make it easy to explain the geometry of lines or the graph of two lines of equations. It contains two variables whose values are unknown.
It contains values of x and y both as a solution to make the two sides of the equation equal.
The linear equations of two variables are often used in the following ways–
The problems can be solved by converting the situation into mathematical statements that tell the relation between the unknown variables. It makes it easier to solve such problems.
It is mainly used in linear programming problems called LPPs which involve analytical thinking and it deals with real-life situations.