Progetto CLIL - Liceo Scientifico “N. Copernico” – a.s. 2016-2017
prof. Silvia Valente
NOTES ON LINEAR EQUATIONS
An equation that can be written in the form ax+by+c=0, where a, b, c are real numbers and a and b
are not both zero a  0  b  0 , is a linear equation in two variables.
ax+by+c=0 is known as the implicit form of the equation of a line.
y=mx+q is called the explicit form or the slope-intercept form of the linear equation (of a non
vertical line).
Its graph is a straight line with slope m and y-intercept ( or intercept on the y-axis) q.
Looking at the equation written in the y-form, the slope of the line is the numerical coefficient of the
x term.
The equation of any line parallel to the y-axis is x=h. The equation of the y-axis is x=0.
The equation of any line parallel to the x-axis is y=k. The equation of the x-axis is y=0.
Slope and y-intercept of a line
y-intercept
q is the y-coordinate of the point where the line crosses the y-axis; a vertical line has no yintercept.
If q>0 the line crosses the y-axis above the origin.
If q<0 the line crosses the y-axis below the origin.
If q=0 the line crosses at the origin.
Slope
The slope of a line (m) is the measure of steepness (of a nonvertical line).
The slope of a line is defined as the ratio of the vertical change to the horizontal change between
any two points of the line. A ratio is the quotient of two numbers.
That is: slope 
vertical change
difference

horizontal change difference
0
m
y
x
of
of
y coordinate s y2  y1

x coordinate s x2  x1
x  0
If m>0 the line rises from left to right, that is a man walking on it from left to right goes uphill. *
If m>0 the angle that the line makes with the positive direction of the x-axis is an acute angle.
If m<0 the line falls, that is a man walking on it from left to right goes downhill.
If m<0 the angle that the line makes with the positive direction of the x-axis is an obtuse angle.
If m=0 the line is parallel to the x-axis. The slope of a horizontal line is 0.
The slope of a vertical line is undefined, that is it doesn’t exist.
If m becomes bigger the line becomes steeper.
Lines with the same m are parallel lines.
Lines with different m have one point of intersection.
*
NB We always read a graph from left to right.
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