# 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{aligned} (-1) = 2(0) -1 \\ (3) = 2(2) -1 \end{aligned} 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{aligned} \mbox{Line 1: } y &= 10x_1 + 5 \\ \mbox{Line 2: } y &= 11x_2 -2 \end{aligned}

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
7 years, 3 months ago

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## Comments

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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?

- 7 years, 3 months ago

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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 $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$.

- 7 years, 3 months ago

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Got it. Thanks.

- 7 years, 2 months ago

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What does line1 or line2 represent in three dimensional space?

- 7 years, 2 months ago

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

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