Question

How do you write $2x+3y+7=3$ in standard form? \[\]

Answer 1

Hint: We recall the three forms of writing a linear equation: the general form $ax+by+c=0$, the slope intercept form $y=mx+c$ and the standard form $ax+by=c$. We see that the given equation $2x+3y+7=3$ has constants on both sides of the equation. We subtract both sides by 7 in order to eliminate constant at the left hand side. \[\]

Complete step by step answer:
We know that a linear equation is an equation with degree 1 with two variables $x,y$ has the general form
\[ax+by+c=0\]
Here $a,b,c$ have to be real numbers and $a,b$ cannot be zero. We know from the Cartesian coordinate system that every linear equation can be represented as a line. If the line is inclined with positive $x-$axis at an angle $\theta $ then its slope is given by $m=\tan \theta $ and if it cuts $y-$axis at a point $\left( 0,c \right)$ from the origin the $y-$intercept is given by $c$. The slope-intercept form of equation is given by
\[y=mx+c\]
We know that the standard form of linear equation otherwise also known as intercept form is written with constant $c$ on the right of equality sign as
\[ax+by=c\]
It is called intercept form too because the $x$ and $y-$are obtained as $\dfrac{c}{a},\dfrac{c}{b}$ at the points$\left( \dfrac{c}{a},0 \right),\left( 0,\dfrac{c}{b} \right)$. We are given in the question the following equation
\[2x+3y+7=3\]
 We see that the constant terms are at both sides of the given equation which are 7 and 3. We subtract 7 both sides to have;
 \[\begin{align}
  & 2x+3y+7-7=3-7 \\
 & \Rightarrow 2x+3y=-4 \\
\end{align}\]
We see that now the constant is at only the right hand side in the above equation and hence it is in standard form. \[\]

Note: We note that we need at least 2 linear equations in two variables to find a unique solution. The standard form of the equation is useful while using elimination methods to solve the linear equations. We can subtract $c$ both sides of the general equation $ax+by+c=0$ to convert it into standard form and add $-c$ both sides of standard equation $ax+b=c$ to convert it into general form.