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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
118
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
shortcut formula.
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
formula.
To nd
the equation
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
point-slope formula,
both would have
the same
equation.
If all the
coefcients have
a common factor,
divide all of the terms
by that number. For
example, if all of the
coefcients are even,
divide everything by 2.
When no
coefcient
is written, you
should assume the
coefcient is 1, so
A = 1.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
119
6.31 Calculate the slope of the line 4x – 9y = 14 using the slope shortcut formula
and verify the answer by writing the equation in slope-intercept form.
Substitute A = 4 and B = –9 into the slope shortcut formula.
Solve the equation 4x – 9y = 14 for y.
The slope of an equation in slope-intercept form is the coefficient of its x-term.
The slope of this line is , which verifies the slope generated by the shortcut
formula.
6.32 What linear equations can be expressed in standard form but cannot be
expressed in slope-intercept form?
Equations of vertical lines cannot be written in slope-intercept form. Vertical
lines have form x = c, where c is the distance between the vertical line and the
y-axis. Because the equations do not contain y, they cannot be solved for yone
major characteristic of lines in point-slope form. Furthermore, vertical lines
have an undefined slope and are, therefore, missing yet another key component
of slope-intercept form.
6.33 Express the equation y – 3 = –7(x – 1) in standard form.
To convert this equation from point-slope form to standard form, begin by
expanding the right side of the equation.
Move –7x left of the equal sign and –3 right of the equal sign by adding those
quantities to both sides of the equation.
This equation is in standard form Ax + By = C because the x- and y-terms are left
of the equal sign, the coefficients and constant share no common factors, no
fractions are present in the equation, and the coefficient of x is positive.
The formula
from
Problem 6.30
Undened
slope” means the
same as “no slope.
Horizontal lines have
a slope of zero, which is
a real number. Vertical
lines dont have a slope
at all. You cant write
an equation in slope-
intercept form if
theres no slope!
The x- and
y-terms also
have to be in the
right order; x has
to come before y,
so the equation
y + 7x = 10—though
equivalent—is not in
standard form.

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