# Equation of a Line

## Definition

A linear equation in two variables, often written as $$y = mx + b$$, describes a line in the $$xy$$-plane. The line consists of all the solution pairs $$(x_0, y_0)$$ that make the equation true.

For example, in the equation, $$y = 2x -1$$, the line would pass through the points $$(0,-1)$$ and $$(2, 3)$$ because these ordered pairs are solutions to the equation, as is every other point on the line.

\begin{align} (-1) = 2(0) -1 \\ (3) = 2(2) -1 \end{align}

A line through the points (0,-1) and (2,3)

The coefficient $$m$$ is the slope of the graph, and the constant $$b$$ is the $$y$$-intercept.

The slope of a graph is the ratio between the rate of change of $$y$$ and the rate of change of $$x$$, so for two points on a graph $$p_1=(x_1,y_1)$$ and $$p_2=(x_2,y_2)$$, slope between them is:

$m = \frac{y_2 - y_1}{x_2-x_1}$

## Technique

### What is the $$y$$-intercept of the line described by $$2x - 3y + 9 = 0$$?

If $$2x - 3y + 9 = 0$$, then $$3y = 2x + 9$$, and so $$y = \frac{2}{3}x + \frac{9}{3}$$, which means the $$y$$-intercept is $$3$$. $$_\square$$

### What is $$a$$ in the point $$(-10, a)$$, if it falls on the line through the points $$(2,4)$$ and $$(3,3)$$?

The slope of the line is $$m = \frac{(3)-(4)}{(3)-(2)}=-1$$, so $$y=-x + b$$ for all $$x$$ and $$y$$ on the graph. This means we can take one of the given points and substitute in its values to find $$b$$.

$(3)=-(3) + b$

Therefore, $$b = 6$$, and the equation of the line is $$y = -x +6$$. This means the point $$(-10, a)$$ satisfies the equation $$a = -(-10) +6$$. Thus, $$a = 16$$. $$_\square$$

## Applications and Extensions

### The intersection of the line $$y = 10x + 5$$ with the line through the points $$(1, 9)$$, $$(2, 20)$$, is a point $$( a , b )$$. Find the sum of $$a$$ and $$b$$.

First, we need to find the equation of the line through $$(1, 9)$$ and $$(2, 20)$$. Since $$\frac{y_2 - y_1}{x_2-x_1} = \frac{20-9}{2-1}$$, the slope of the line must be 11. Further, since $$9 = 11(1) + b$$, the $$y$$-intercept is $$-2$$.

Thus, our two equations are: \begin{align} \mbox{Line 1: } y &= 10x_1 + 5 \\ \mbox{Line 2: } y &= 11x_2 -2 \end{align}

To find the point of intersection $$(a,b)$$, we solve the system of equations above by equating the two $$y$$'s.

$10x + 5 = 11 x -2$

Therefore, $$x = 7$$, and placing $$7$$ into either of the equations for $$x$$ gives us $$(7, 75 )$$ as the point of intersection. The answer, then, is $$7+75 = 82$$. $$_\square$$

Note by Arron Kau
4 years, 1 month ago

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I keep wondering at times, why is slope $$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ and not $$m = \frac{x_{2} - x_{1}}{y_{2} - y_{1}}$$ ? Both of them give same kind of information about the slope, one indicates the change in y per change in x and the other indicates the change in x per change in y. Is it just convention or is there some other reason to it?

- 4 years, 1 month ago

well since $$m=\tan x$$, its very obvious that $$m=\tan x=\frac{y_2-y_1}{x_2-x_1}$$. Stay in this pattern is also very good because in higher grades we may need to solve like "What is the equation of the line which is perpendicular with line $$5x+2=0$$ and pass through the point $$(7,0)$$?". Keep it $$m=\tan x$$ rather than $$\cot x$$ is more simple.

Or another way to explain is as a function $$f(x)$$, $$x$$ is the one who "change" and $$y$$ is the one who "change because $$x$$ changed". Then as $$m$$ explains the different rate of change of $$y$$ against $$x$$, then $$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$.

- 4 years, 1 month ago

Got it. Thanks.

- 4 years, 1 month ago

What does line1 or line2 represent in three dimensional space?

- 4 years ago

They represent two lines, since you can write them as: $0z+y=10x_1+5 \\ 0z+y=11x_2-2.$

- 3 years, 11 months ago