Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
Multiply the numerator and denominator by the reciprocal of the denominator.
The line connecting A and C has the same slope as the line connecting A and B;
it also shares a common point, A. Therefore, the lines are not unique. Instead,
points A, B, and C all belong to the same line.
Standard Form of a Linear Equation
Write equations of lines in a uniform way
6.29 What are the characteristics of a linear equation in standard form?
The standard form of a linear equation is Ax + By = C. In standard form, the
variable terms are located left of the equal sign, and the constant is located
right of the equal sign. There are additional restrictions on the constants.
For one thing, A, B, and C cannot share common factors (other than 1). The
coefﬁcients must all be integers, which means that no fractions are present in
the equation. Finally, A, the coefﬁcient of the x-term, must be nonnegative. If A
is negative, multiply each term of the equation by –1.
6.30 According to Problem 5.37, the slope of Ax + By = C, a linear equation in stan-
dard form, is . Use this shortcut formula to calculate the slope of
x + 4y = –3, and verify the answer by writing the equation in slope-intercept form.
The equation x + 4y = –3 is in standard form, so the coefﬁcients of x and y are
A and B, respectively: A = 1 and B = 4. Substitute these values into the slope
Solve the equation x + 4y = –3 for y to express it in slope-intercept form.
The slope of an equation in slope-intercept form is the coefﬁcient of the x-term.
The slope of this line is , which veriﬁes the slope generated by the shortcut
of a line, you
need a point and a
slope. Both of these
slopes are equal and
both lines pass through
B, so according to the
both would have
If all the
a common factor,
divide all of the terms
by that number. For
example, if all of the
coefcients are even,
divide everything by 2.
is written, you
should assume the
coefcient is 1, so
A = 1.