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Chapter 4: Problem 13

Complete the ordered pair for each equation. $$y=-x+6 ;(9,)$$

Short Answer

Expert verified
The completed ordered pair for the equation \(y=-x+6\) with the given x-coordinate (9) is \((9, -3)\).

Step by step solution

01

Write down the equation of the line

The equation of the line is given as \(y=-x+6\).
02

Write down the given ordered pair

The given ordered pair with the missing value is \((9,)\).
03

Substitute the x-coordinate in the equation

Replace \(x\) with the given value of 9. So, the equation becomes: \(y = -9 + 6\).
04

Solve for the y-coordinate

Perform the necessary operation to find the value of \(y\). In our case: \(y = -9 + 6 \\ y = -3\)
05

Write down the completed ordered pair

Since the x-coordinate is 9 and the y-coordinate is -3, the ordered pair \((x, y)\) is \((9, -3)\).

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Key Concepts

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

Understanding Ordered Pairs
Ordered pairs are a fundamental concept in algebra and coordinate geometry. These pairs, written as \((x, y)\), represent points on the coordinate plane. Here's why they're important:
  • Identification: The first number in an ordered pair, known as the x-coordinate, specifies the horizontal position of a point. The second number, the y-coordinate, determines the vertical position.
  • Mapping: Each ordered pair corresponds uniquely to a position in the coordinate plane.
When given an ordered pair like \((9, )\), the goal is to find the missing value using an equation. In this scenario, the x-coordinate is provided (9), and we need to solve for the missing y-coordinate using the linear equation provided.
Ordered pairs are essential tools for visualizing points on graphs and for solving more complex mathematical problems.
Basics of Linear Equations
Linear equations are equations of the first degree, meaning they involve only the first power of the variable. The standard form is:\[y = mx + b\]
  • m: This represents the slope of the line. It indicates how steep the line is.
  • b: This is the y-intercept. It shows where the line crosses the y-axis.
The linear equation given in our exercise, \(y = -x + 6\), is in slope-intercept form. Here, it can be interpreted as having a slope of -1 (meaning the line descends) and intercepting the y-axis at 6.
Linear equations are among the most common types of equations you will encounter. They are straightforward to graph, and they always produce straight lines on a coordinate plane.
Understanding these basics will help in solving equations by recognizing how the components fit together to form a graph.
Solving Equations
Solving equations is a critical skill in algebra. It involves finding the values of the variables that make the equation true. Here's a simple approach:
  • Substitute: Place the given values into the equation. For our ordered pair, substitute the x-value into the equation.
  • Simplify: Perform algebraic operations to simplify the equation. This involves basic arithmetic and following the order of operations.
Using the original example \(y=-x+6\), by substituting \(x = 9\), we derive \(y = -9 + 6\). Simplifying this gives \(y = -3\).
This final value completes the ordered pair, giving us \((9, -3)\).
Mastering these steps will enable you to tackle a wide variety of algebra problems confidently and accurately.

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Most popular questions from this chapter

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((0,2)\) and \((6,0)\)

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(-3 x+4 y=8\) containing \((7,4)\)

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. \(y=x ;(10,-4) ;\) slope-intercept form

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. $$y=\frac{2}{3} x+4 ;(6,-3) ; \text { slope-intercept form }$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=2 x+\frac{1}{2}$$
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