Equation of a Straight Line Flashcards

All 6 cards in this deck

What is the equation of a straight line with gradient $m$ passing through the point $(x_1,y_1)$?
e.g. gradient $3$ through $(2,5)$

$y - y_1 = m(x - x_1)$
e.g. $y - 5 = 3(x - 2)$

What are the steps to find the equation of the line through two given points?
e.g. $(1,2)$ and $(3,8)$

1. Gradient: $m = \dfrac{8-2}{3-1} = 3$
2. Use $y - y_1 = m(x - x_1)$: $y - 2 = 3(x - 1)$
3. Rearrange: $y = 3x - 1$

What are the steps to find the gradient and $y$-intercept of a line given in the form $ax + by + c = 0$?
e.g. $2x + 5y - 10 = 0$

1. Make the $y$ term the subject: $5y = -2x + 10$
2. Divide by the coefficient of $y$: $y = -\tfrac25 x + 2$
3. Read off: gradient $-\tfrac25$, $y$-intercept $2$

What are the steps to find the equation of the line perpendicular to a given line through a given point?
e.g. perpendicular to $y = 2x + 1$ through $(4,3)$

1. Gradient of the given line: $2$
2. Negative reciprocal: $-\tfrac12$
3. Use $y - y_1 = m(x - x_1)$: $y - 3 = -\tfrac12(x - 4)$, i.e. $y = -\tfrac12 x + 5$

What are the steps to find the point of intersection of two straight lines algebraically?
e.g. $y = 2x + 1$ and $y = x + 4$

1. Set the expressions equal: $2x + 1 = x + 4$
2. Solve: $x = 3$
3. Substitute back: $y = 7$, so the point is $(3,7)$

What are the steps to find where a straight line crosses the axes?
e.g. $y = -2x + 6$

1. Put $x = 0$ for the $y$-axis: $(0,6)$
2. Put $y = 0$ for the $x$-axis: $0 = -2x + 6$, so $(3,0)$