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

coefcients have

a common factor,

divide all of the terms

by that number. For

example, if all of the

coefcients are even,

divide everything by 2.

When no

coefcient

is written, you

should assume the

coefcient 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 y—one

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

“Undened

slope” means the

same as “no slope.”

Horizontal lines have

a slope of zero, which is

a real number. Vertical

lines don’t have a slope

at all. You can’t write

an equation in slope-

intercept form if

there’s 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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