Linear equations in two variables and functions

Rectangular coordinate system

We draw graphs of data by using two intersecting perpendicular lines called rectangular coordinate system. The horizontal and the vertical lines are called $x$-axis and $y$-axis respectively.The point, where the two axes intersect is called the origin. At the origin, both the $x$ and $y$ values are 0. On the $x$-axis, the numbers to the right of the $y$ axis are positive and that to the left are negative. On the $y$ axis, numbers above the $x$-axis is positive and the numbers below are negative. The axes divide the plane of the graphing area into four regions called quadrants.

Linear equations in two variables

Standard form of a linear equation of two variable is

$Ax+By=C$

where $x$ and $y$ are the variables and $A$,$B$ and $C$ are the real numbers.

The solution of the equation is written as ordered pair: $(x,y)$.

$x$ and $y$-intercepts

$x$-intercept is a point where a graph intersects the $x$ axis. Since at the $x$ intercept, $y=0$, an $x$-intercept is in the form $(a,0)$.

To find the $x$-intercept put $y=0$ in the equation and solve for $x$, if the solution is $a$, the $x$ intercept is $(a,0)$.

$y$-intercept is a point where a graph intersects the $y$ axis. Since at the $y$ intercept, $y=0$, an $y$-intercept is in the form $(0,b)$.

To find the $y$-intercept put $x=0$ in the equation and solve for $y$, if the solution is $b$, the $y$ intercept is $(0,b)$.

Horizontal and vertical line

The graph of the equation $y=k$, where $k$ is a constant, is a horizontal line and

The graph of the equation $x=k$, where $k$ is a constant, is a vertical line

Slope of a line

Slope of a line is the ratio of the change in $y$ to the change in $x$. i.e., rise over run.

$slope =\dfrac{change \:in\: y}{change\: in\: x}$$=\dfrac{rise}{run}$

To find the slope of a line, we need two points. If a line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$, then the slope of the line is

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

Parallel and perpendicular lines

If two lines are on a plane and they do not intersect, they are parallel.

Slope of parallel lines are the same. i.e., if $m_1$ and $m_2$ are the slopes of two parallel lines, then

$m_1=m_2$

If two lines intersect at a right angle, they are perpendicular lines.

If $m_1$ and $m_2$ are the slopes of two perpendicular lines, then

$m_1 \,m_2=-1$

or

$m_1=-\dfrac{1}{m_2}$

Interpretation of slope and applications

Equations of a line

The standard form of an equation of a line is

$Ax+By=C$

There are other two forms: slope-intercept form and point-slope form.

Point-slope form

An equation of the form:

$y=mx+b$

is the slope-intercept form.

where $m$ is the slope of the line and $(0,b)$ is the $y$-intercept of the line.

Point-slope form

An equation of a line of the form:

$y-y_1=m(x-x_1)$

is the point-slope form.