Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews March 31, 2023March 24, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

An ordered pair is a pair of numbers in a specific order. For example, (1, 2) and (- 4, 12) are ordered pairs. The order of the two numbers is important: (1, 2) is not equivalent to (2, 1) -- (1, 2)≠(2, 1).

Using Ordered Pairs to Represent Variables

Ordered pairs are often used to represent two variables. When we write (x, y) = (7, - 2), we mean x = 7 and y = - 2. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.

Example 1. If (x, y) = (- 1, 4), what is the value of 3x + 2y - 4 ? 3x + 2y - 4 = 3(- 1) + 2(4) - 4 = - 3 + 8 - 4 = 1

Example 2. Which of the following ordered pairs (x, y) are solutions to the equation - 6 = 1 ? {(4, 1),(5, 2),(- 3, 1),(- 3, -1),(1, 4)} (x, y) = (4, 1): -6 = - 6 = 7 - 6 = 1. Solution. (x, y) = (5, 2): -6 = -6 = -6 = - ≠1. Not a solution. (x, y) = (- 3, 1): -6 = -6 = - 7 - 6 = - 13≠1. Not a solution. (x, y) = (- 3, - 1): -6 = - 6 = 7 - 6 = 1. Solution. (x, y) = (1, 4): -6 = -6 = -6 = - ≠1. Not a solution. Thus, {(4, 1),(- 3, -1)} are solutions to - 6 = 1.

Graphing Ordered Pairs

We have graphed values on the number line in pre- algebra and in earlier chapters of algebra. However, we can only graph points of one variable on the number line; thus, we need a 2-dimensional (2 variable) way of representing points -- the xy-graph:

The horizontal axis, called the x-axis, represents values of x, and the vertical axis, called the y-axis, represents values of y. From now on, the word "graph" will refer to the xy-graph, unless specified otherwise.

To graph a point on the xy-graph, first find the x-coordinate on the x-axis. Then move up on the graph the number of spaces which is equal to the y-coordinate (or move down if the y-coordinate is negative). For example, to graph (2, 3), find 2 on the x-axis. Then move up 3 spaces. To graph (- 2, 1), find -2 on the x-axis, then move up 1 space. To graph (1.5, - 1), find 1.5 on the x-axis, then move down1 space: