Linear Equations
Linear Equations are equations that typically define the relationship between two variables
(such as x and y) with the use of an arithmetic equation that does not include exponents or
radicals. They are called linear equations as they form a straight line when graphed. The
table below shows some important terms and examples, but a linear equation will always
have the form y = mx+c, where y and x are two related variables, m is the gradient,
measured by change in value of y divided by change in value of x, and c is the value of y
when x is zero (the y-intercept).
Defining and Forming Linear Equations
Linear Equation General Form Example
Slope intercept form y = mx + b y = 2x + 3
Point–slope form y – y1 = m(x – x1 ) y – 3 = 6(x – 2)
General Form Ax + By + C = 0 2x + 3y – 6 = 0
Intercept form x/x0 + y/y0 = 1 x/2 + y/3 = 1
As a Function f(x) instead of y f(x) = x + 3
f(x) = x + C
The Identity Function f(x) = x f(x) = 3x
Constant Functions f(x) = C f(x) = 6
The y Rise/run is the gradient. When it's given that If the coordinates are
intercept is The gradients of parallel the line goes through given as variables, use
always at lines are equal, while the origin, you can them as you would
(0,b). Thus those of perpendicular often use (0, 0) as a numbers to find the
y=b when lines are negative point to help solve slope or other values of
x=0. reciprocals of each the problem. a line.
other.
How do we build linear equations?
When I bought a car it had 27,000 miles on it. Every year, I drive it about 12,000 more miles.
Using this information, we can set up a linear equation, defining y as the miles on it, and x as
the time in years after buying it. y = 12,000x + 27,000, because I drive it 12,000 more miles
every year (meaning change in y over change in x), and that's our rate of increase, i.e. the
SAT with Talal 1
, gradient. And when x = 0 (I just bought the car), the miles were 27,000, which is our
y-intercept.
In addition to slope, you also need to know what x and y intercepts are. The x-intercept is
where the graph crosses the r-axis. Likewise, the y-intercept is where the graph crosses the
y-axis.
Let's say we have the line: 2x + 3y = 12
To find the x-intercept, set y equal to 0
2x + 3(0) = 12
2x = 12
x=6
The x-intercept is 6.
To find the y-intercept, set x equal to 0. Then let’s solve it, as the line naturally passes
through the y-axis when the corresponding value of x = 0.
2(0) +3y = 12
3y = 12
y=4
The y-intercept is 4.
All lines can be expressed in slope-intercept form: y = mx + b, where m is the slope and b is
the y-intercept. So for the line y = 2x - 3, the slope is 2 and the y-intercept is -3.
While all lines can be expressed in slope-intercept form, sometimes it'll take some work to
get there. If you're given a slope and a y-intercept then it's really easy to get the equation of
the line.
But what if we're handed a slope and a point instead of a slope and a y-intercept? Then it'll
be more convenient to use point-slope form: y-y1 = m(x-x1)
where (x1, y1) is the given point.
For example, let's say we want to find the equation of a line that has a slope of 3 and passes
through the point (1, -2). The equation of the line is then
y - (-2) = 3(x - 1)
SAT with Talal 2
Linear Equations are equations that typically define the relationship between two variables
(such as x and y) with the use of an arithmetic equation that does not include exponents or
radicals. They are called linear equations as they form a straight line when graphed. The
table below shows some important terms and examples, but a linear equation will always
have the form y = mx+c, where y and x are two related variables, m is the gradient,
measured by change in value of y divided by change in value of x, and c is the value of y
when x is zero (the y-intercept).
Defining and Forming Linear Equations
Linear Equation General Form Example
Slope intercept form y = mx + b y = 2x + 3
Point–slope form y – y1 = m(x – x1 ) y – 3 = 6(x – 2)
General Form Ax + By + C = 0 2x + 3y – 6 = 0
Intercept form x/x0 + y/y0 = 1 x/2 + y/3 = 1
As a Function f(x) instead of y f(x) = x + 3
f(x) = x + C
The Identity Function f(x) = x f(x) = 3x
Constant Functions f(x) = C f(x) = 6
The y Rise/run is the gradient. When it's given that If the coordinates are
intercept is The gradients of parallel the line goes through given as variables, use
always at lines are equal, while the origin, you can them as you would
(0,b). Thus those of perpendicular often use (0, 0) as a numbers to find the
y=b when lines are negative point to help solve slope or other values of
x=0. reciprocals of each the problem. a line.
other.
How do we build linear equations?
When I bought a car it had 27,000 miles on it. Every year, I drive it about 12,000 more miles.
Using this information, we can set up a linear equation, defining y as the miles on it, and x as
the time in years after buying it. y = 12,000x + 27,000, because I drive it 12,000 more miles
every year (meaning change in y over change in x), and that's our rate of increase, i.e. the
SAT with Talal 1
, gradient. And when x = 0 (I just bought the car), the miles were 27,000, which is our
y-intercept.
In addition to slope, you also need to know what x and y intercepts are. The x-intercept is
where the graph crosses the r-axis. Likewise, the y-intercept is where the graph crosses the
y-axis.
Let's say we have the line: 2x + 3y = 12
To find the x-intercept, set y equal to 0
2x + 3(0) = 12
2x = 12
x=6
The x-intercept is 6.
To find the y-intercept, set x equal to 0. Then let’s solve it, as the line naturally passes
through the y-axis when the corresponding value of x = 0.
2(0) +3y = 12
3y = 12
y=4
The y-intercept is 4.
All lines can be expressed in slope-intercept form: y = mx + b, where m is the slope and b is
the y-intercept. So for the line y = 2x - 3, the slope is 2 and the y-intercept is -3.
While all lines can be expressed in slope-intercept form, sometimes it'll take some work to
get there. If you're given a slope and a y-intercept then it's really easy to get the equation of
the line.
But what if we're handed a slope and a point instead of a slope and a y-intercept? Then it'll
be more convenient to use point-slope form: y-y1 = m(x-x1)
where (x1, y1) is the given point.
For example, let's say we want to find the equation of a line that has a slope of 3 and passes
through the point (1, -2). The equation of the line is then
y - (-2) = 3(x - 1)
SAT with Talal 2
