Equation of a straight line in point-gradient form
To write the equation of a straight line you do not always have its y-intercept to hand. Sometimes you only know one point that the line passes through and its gradient. The point-gradient form is made for exactly this: building the equation without having to work out first where the line crosses the $Y$ axis.
The point-gradient equation
Write the equation of the line in point-slope form that passes through the point $A(3, -2)$ and has a gradient of $m = 4$.
Write the equation of the line in point-slope form that passes through the point $A(3, -2)$ and has a gradient of $m = 4$.
Calculate the gradient of the line passing through the points $P(-1, 4)$ and $Q(2, 10)$ and write its equation in point-slope form using the coordinates of point $P$.
Calculate the gradient of the line passing through the points $P(-1, 4)$ and $Q(2, 10)$ and write its equation in point-slope form using the coordinates of point $P$.
From point-gradient form to explicit form
To go from $y - y_1 = m(x - x_1)$ to the explicit form $y = mx + n$, you just have to work it out: expand the bracket and make $y$ the subject.
Given the line with equation $y - 5 = -2(x + 1)$, state an exact point it passes through and its gradient. Then, rearrange the expression to write it in explicit form.
Given the line with equation $y - 5 = -2(x + 1)$, state an exact point it passes through and its gradient. Then, rearrange the expression to write it in explicit form.
A line has the point-slope equation $y + 3 = \frac{1}{2}(x - 6)$. Check mathematically whether the point $C(8, -2)$ lies on this line.
A line has the point-slope equation $y + 3 = \frac{1}{2}(x - 6)$. Check mathematically whether the point $C(8, -2)$ lies on this line.
Parallel and perpendicular lines
Write the equation in point-slope form of the line parallel to $y = 3x - 7$ passing through the point $B(-4, 1)$.
Write the equation in point-slope form of the line parallel to $y = 3x - 7$ passing through the point $B(-4, 1)$.
Find the equation in point-slope form of the line that is perpendicular to $y = -\frac{1}{3}x + 5$ and passes through the point of intersection of the lines $y = 2x - 1$…
Find the equation in point-slope form of the line that is perpendicular to $y = -\frac{1}{3}x + 5$ and passes through the point of intersection of the lines $y = 2x - 1$ and $y = x + 3$.
Lines with a geometric meaning: perpendicular bisector and altitudes
The point-gradient form is very useful when the line you are looking for is defined by a geometric condition: it passes through a midpoint, it is perpendicular to a side, and so on. The procedure is always the same: find a point that the line passes through and its gradient, and apply the formula.
Write the equation in point-slope form of the perpendicular bisector of the segment with endpoints $A(-2, 5)$ and $B(4, 1)$. Remember that the perpendicular bisector is…
Write the equation in point-slope form of the perpendicular bisector of the segment with endpoints $A(-2, 5)$ and $B(4, 1)$. Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint.
A van hire company charges a fixed amount plus a constant rate per kilometre driven. If for a journey of $50$ km they charge $65$ € and for one of $120$ km they charge…
A van hire company charges a fixed amount plus a constant rate per kilometre driven. If for a journey of $50$ km they charge $65$ € and for one of $120$ km they charge $107$ €, determine the gradient of the cost function and write its equation in point-slope form.
In the triangle with vertices $A(-2, -1)$, $B(4, 1)$ and $C(1, 5)$, find the equation in point-slope form of the line containing the altitude corresponding to vertex $C$.
In the triangle with vertices $A(-2, -1)$, $B(4, 1)$ and $C(1, 5)$, find the equation in point-slope form of the line containing the altitude corresponding to vertex $C$.