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Chapter 4: Problem 16

Complete the ordered pair for each equation. $$-x+12 y=7 ;\left(, \frac{3}{4}\right)$$

Short Answer

Expert verified
The complete ordered pair for the given equation \(-x + 12y = 7\) and the given y-coordinate \(\frac{3}{4}\) is \(\left(2, \frac{3}{4}\right)\).

Step by step solution

01

Identify the given information

We are given a linear equation, \(-x + 12y = 7\), and an ordered pair with a missing x-coordinate, \(\left(, \frac{3}{4}\right)\).
02

Substitute the given y-coordinate into the equation

Replace y in the equation with the given y-coordinate, \(\frac{3}{4}\): \(-x + 12\left(\frac{3}{4}\right) = 7\)
03

Simplify and solve for x

Next, we need to simplify the equation and solve for x: \(-x + 12\left(\frac{3}{4}\right) = -x + 9 = 7\) Now, isolate x by adding x to both sides and subtracting 7 from both sides: \(-x + x + 9 = x + 7\) \(9 = x + 7\) Now, subtract 7 from both sides: \(9-7=x\) So, the x-coordinate is: \(x=2\)
04

Write the complete ordered pair

Now we can write the complete ordered pair with the x and y coordinates: \(\left(2, \frac{3}{4}\right)\)

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Key Concepts

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

Ordered Pairs
An ordered pair consists of two numbers that define a point in a plane. The first number, usually represented by \(x\), specifies the horizontal position, while the second number, \(y\), specifies the vertical position.
The format of an ordered pair is \((x, y)\). For example, if you see \((2, \frac{3}{4})\), this describes a point on the coordinate plane. The value of \(x\) is 2, indicating how far you move along the horizontal axis, and \(\frac{3}{4}\) tells how far you move up or down on the vertical axis.
Ordered pairs are crucial for locating points precisely on a graph and are often used in conjunction with equations to understand geometrical shapes or paths.
Coordinate Geometry
Coordinate geometry, or Cartesian geometry, is a branch of mathematics that allows us to specify the exact position of a point in space using numerical coordinates. These coordinates are defined in reference to two perpendicular intersecting lines called axes (x-axis and y-axis).
This system helps in graphing equations, finding line slopes, and determining intersections of lines. By using ordered pairs, we can depict points, lines, and curves in a visually structured layout.
  • The x-coordinate indicates how far a point is along the horizontal axis.
  • The y-coordinate specifies how far a point is along the vertical axis.
Coordinate geometry provides the tools to translate algebraic equations into visual representations, which makes understanding complex algebraic expressions simpler.
Solving Equations
Solving equations is a fundamental aspect of mathematics. It involves finding the value(s) of unknown variables that make an equation true. When given an equation, such as \(-x + 12y = 7\), and part of an ordered pair \(( , \frac{3}{4})\), you can substitute the known value of \(y = \frac{3}{4}\) into the equation to find \(x\).
To solve for \(x\):
  • Substitute the known \(y\)-value into the equation.
  • Simplify the equation to bring common terms together.
  • Isolate the variable \(x\) by reversing operations, ensuring the equation remains balanced.
Through these steps, we find that: 1. Plugging in gives us \(-x + 9 = 7\).
2. Solving for \(x\), we discover \(x = 2\).
Hence, the complete ordered pair is written \((2, \frac{3}{4})\). Solving equations allows us to deduce the missing information and understand relationships between variables.

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Most popular questions from this chapter

Graph each function by finding the \(x\) - and \(y\) -intercepts and one other point. $$f(x)=-x$$

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. \(x+y=9 ;(-5,-5) ;\) slope-intercept form

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=2\) and \(y\) -intercept \((0,-11)\)

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. $$24 x-15 y=10 ;(16,-7) ; \text { standard form }$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{12}{x-11}$$
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