Equation of a linear relation

There is more than one form of a linear relation. The most common form is called the slope-intercept form, which is

\[y=mx+b\]

In this form,

\(x\) is the variable on the horizontal axis;
\(y\) is the variable on the vertical axis;
\(m\) is the slope of the line; and
\(b\) is the \(y\)-intercept of the line.

Another form is the standard form, which is

\[Ax+By+C=0\]

In this form, \(A,B\) and \(C\) are integer values, and \(A\) is a positive number. A linear equation can be changed from one form to another, yet it remains the same linear equation and is graphed as one and the same line.

An equation represents a linear relation if and only if all the variables in the relation have an exponent of \(\boldsymbol{1}\). If a variable has any other exponent, the relation is not linear. All the following relations are linear, and the equations can be re-arranged into the slope-intercept form.

\[\begin{align} 8x+4y &= 20;\qquad & \frac{1}{2}x+\frac{2}{3}y &= \frac{1}{6};\qquad & \frac{2}{3}y-2x-1 &= 0 \\ 2x+y &= 5;\qquad & 3x+4y &= 1;\qquad & 2y-6x-3 &= 0 \\ y &= -2x+5;\qquad & 4y &= -3x+1;\qquad & 2y &= 6x+3 \\ & & y &= -\frac{3}{4}x+\frac{1}{4};\qquad & y &= 3x+\frac{3}{2} \end{align}\]

Finding the equation of a linear relation

In order to find the equation of a line we need

the slope of the line and one point on the line; or
two points on the line.

With the slope and a point

If the known point on the line is the \(y\)-intercept, we can immediately write the equation. For instance, with a slope of \(2\) and a \(y\)-intercept of 3, the equation is

\[y=2x+3\]

If the point is not the \(y\)-intercept, we substitute with the coordinates of the point in the equation of the line to find the \(y\)-intercept (which is the \(b\)-value). For instance, if the line has a slope of \(-3\) and passes through the point \((1,-1)\), the equation is initially

\[y=-3x+b\]

If we substitute with the coordinates \(x=1\) and \(y=-1\), we get

\begin{align} -1 &= (-3)(1)+b \\ -1 &= -3 + b \\ b &= 2 \end{align}

Therefore, the equation of the line is

\[y=-3x+2\]

With two points

We can calculate the slope of the line from the two points, then we can choose one of them and proceed as discussed above. For instance, to find the equation of the line that passes through \((-2,1)\) and \((1,4)\), we calculate the slope as follows

\[m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{4-1}{1-(-2)}=\dfrac{3}{3}=1\]

Therefore, the equation of the line is initially

\[y=x+b\]

Let us substitute with the coordinates \(x=1\) and \(y=4\). We then get

\begin{align} 4 &= 1 + b \\ b &= 3 \end{align}

Therefore, the equation of the line is

\[y=x+3\]