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Chapter 3: Problem 5

The ordered pair \((4,_____)\) is a solution of the equation \(y=3\).

Short Answer

Expert verified
The ordered pair is (4, 3).

Step by step solution

01

Understand What the Equation Represents

Identify that the equation given is a horizontal line, where for any value of x, the value of y is always 3. This means the y-coordinate is consistently 3 regardless of the x-coordinate.
02

Identify the Missing Element

An ordered pair describes a specific point on the plane with an x-coordinate and a y-coordinate. Given that the y-coordinate must satisfy the equation, substitute y in the ordered pair \(4, \_\_\) with 3.
03

Form the Complete Ordered Pair

Replace the blank in the ordered pair \(4, \_\_\) with the value 3. Therefore, the ordered pair becomes \(4, 3\).\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Lines
In mathematics, a horizontal line is a line that runs from left to right and is always parallel to the x-axis of the coordinate plane. This means that the value of y remains constant for all points on the line. For example, the equation \( y = 3 \) describes a horizontal line where y is always 3, no matter what the x-value is.

Horizontal lines are straightforward and have one main characteristic: the y-coordinate does not change. This makes them very simple to understand as the slope鈥攐ften symbolized as 'm' in the slope-intercept form equation \( y = mx + b \)鈥攊s zero. The slope being zero means there is no vertical change as you move along the line.

To sum up:
  • Horizontal lines run left to right.
  • These lines have a constant y-value.
  • The equation \( y = c \) (where c is a constant) represents a horizontal line.
Y-Coordinate
The y-coordinate in an ordered pair \((x, y)\) specifies the vertical position of a point on the coordinate plane. It tells us how far up or down the point is from the x-axis.

In the context of the given exercise, the y-coordinate for any point on the line represented by \( y = 3 \) is always 3. This means that for the ordered pair \((4, \textunderscore\textunderscore)\), the y-coordinate must be 3 for it to be a solution.

It's important to remember that:
  • The y-coordinate tells us the vertical position.
  • In our example, it is fixed at 3 because of the equation \( y = 3 \).
  • This vertical consistency means the point (4, 3) lies on the line.
Equation Solution
Solving for ordered pairs in an equation like \( y = 3 \) means finding points that satisfy the equation. Each point is in the form of \((x, y)\).

In the exercise given, we start with the ordered pair \((4, \textunderscore\textunderscore)\) and need to determine the y-coordinate that makes this pair a solution.

Steps for determining the equation solution:
  • Identify the type of line represented by the equation. Here, it is a horizontal line.
  • Understand that for \( y = 3 \), the y-coordinate must be 3 for any x-value.
  • Substitute the missing y-coordinate with 3 to get the complete ordered pair \((4, 3)\).


By following these steps, you can solve for any ordered pair given a specific line equation, thus understanding how points relate to equations in coordinate geometry.

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Most popular questions from this chapter

\(-4 x+3 y=4\) \(-8 x+6 y=0\)

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 4 x-5 y=0 $$

The table lists the average annual cost \(y\) (in dollars) of tuition and fees at 2 -year colleges for selected years \(x,\) where year 1 represents 2012 , year 2 represents 2013 , and so on. $$ \begin{aligned} &\begin{array}{|c|c|c|} \hline \text { Year } & x & \text { cost (in dollars) } \\ 2012 & 1 & 3310 \\ 2013 & 2 & 3340 \\ 2014 & 3 & 3370 \\ 2015 & 4 & 3460 \\ 2016 & 5 & 3520 \\ \hline \end{array}\\\ &\text { Data from The College Board. } \end{aligned} $$ (a) Write five ordered pairs \((x, y)\) for the data. (b) Plot the ordered pairs \((x, y)\). Do the points lie approximately in a straight line? (c) Use the ordered pairs (2,3340) and (4,3460) to write an equation of a line that approximates the data. Give the final equation in slope-intercept form. (d) Use the equation from part (c) to estimate the average annual cost at 2 -year colleges in 2017 to the nearest dollar. (Hint: What is the value of \(x\) for \(2017 ?)\)

Graph each linear equation. $$ y=-\frac{3}{4} x+3 $$

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ -3 x+2 y=12 $$
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