Calculadora de líneas

Your input: find the equation of a line given two points $P=\left(-4, 7\right)$ and $Q=\left(1, 2\right)$.

The slope of a line passing through the two points $\displaystyle{\large{{P}={\left({x}_{{1}},{y}_{{1}}\right)}}}$ and $\displaystyle{\large{{Q}={\left({x}_{{2}},{y}_{{2}}\right)}}}$ is given by $\displaystyle{\large{{m}=\frac{{{y}_{{2}}-{y}_{{1}}}}{{{x}_{{2}}-{x}_{{1}}}}}}$.

We have that $x_1=-4$, $y_1=7$, $x_2=1$, $y_2=2$.

Plug the given values into the formula for slope: $m=\frac{\left(2\right)-\left(7\right)}{\left(1\right)-\left(-4\right)}=\frac{-5}{5}=-1$.

Now, the y-intercept is $\displaystyle{\large{{b}={y}_{{1}}-{m}\cdot{x}_{{1}}}}$ (or $\displaystyle{\large{{b}={y}_{{2}}-{m}\cdot{x}_{{2}}}}$, the result is the same).

$b=7-\left(-1\right) \cdot \left(-4\right)=3$.

Finally, the equation of the line can be written in the form $\displaystyle{\large{{y}={m}{x}+{b}}}$.

$y=-x+3$.

Answer:

The slope of the line is $m=-1$.

The equation of the line in the slope-intercept form is $y=-x+3$.

The equation of the line in the point-slope form is $y - 7 = - (x + 4)$.

The equation of the line in the point-slope form is $y - 2 = - (x - 1)$.

The general equation of the line is $x + y - 3 = 0$.