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Chapter 8: Problem 36

Complete each ordered pair solution of the given equations. $$ y=-x ;(, 0),(2, \quad),(, 2) $$

Short Answer

Expert verified
The completed pairs are (0,0), (2,-2), (-2,2).

Step by step solution

01

Understand the Equation

The equation given is \( y = -x \). This equation means that for any value of \( x \), \( y \) will be the negative of that value.
02

Solve for the first ordered pair \((?,0)\)

We need to find the value of \( x \) when \( y = 0 \). According to the equation, \( 0 = -x \). Solving for \( x \), we get \( x = 0 \). Thus, the first ordered pair is \((0,0)\).
03

Solve for the second ordered pair \((2, ?)\)

Substitute \( x = 2 \) into the equation \( y = -x \). Thus, \( y = -2 \). This means the second ordered pair is \((2, -2)\).
04

Solve for the third ordered pair \((?,2)\)

We need to find \( x \) when \( y = 2 \). Substitute \( y = 2 \) into the equation \( y = -x \), giving us \( 2 = -x \). Solving for \( x \), we find \( x = -2 \). So, the third ordered pair is \((-2,2)\).

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Key Concepts

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

Linear Equations
A linear equation is an equation that creates a straight line when graphed. It typically has two variables: usually denoted by \( x \) and \( y \). The equation given in your exercise is \( y = -x \), which is a simple type of linear equation.
For linear equations:
  • The graph of the equation will always be a line.
  • Each solution forms an ordered pair \((x, y)\).
  • The equation can always be written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In your equation, \( y = -x \), \( m = -1 \) and \( b = 0 \).
This means every increase in \( x \) yields an equal decrease in \( y \). The solutions \((0, 0)\), \((2, -2)\), and \((-2, 2)\) satisfy the given equation and represent specific points on a straight line.
Coordinate Plane
Understanding the coordinate plane is crucial in solving equations graphically. It consists of two axes:
  • The x-axis is horizontal.
  • The y-axis is vertical.
The point where the axes intersect is known as the origin \((0,0)\). The coordinate plane is divided into four quadrants. Each quadrant has different signs for the ordered pairs:
  • Quadrant I: \((+,+)\)
  • Quadrant II: \((-,+)\)
  • Quadrant III: \((-,-)\)
  • Quadrant IV: \((+,-)\)
In your problem, the ordered pairs like \((0, 0)\), \((2, -2)\), and \((-2, 2)\) lie on this plane. They can help illustrate how different values of \( x \) and \( y \) interact along the line created by the equation.
Negative Slope
The slope of a line describes its steepness and direction. A negative slope implies that as the value of \( x \) increases, the value of \( y \) decreases. This is evident in the equation \( y = -x \), where each increase in \( x \) corresponds to a decrease in \( y \).
When working with linear equations:
  • A positive slope means the line rises from left to right.
  • A negative slope, like in your equation, means the line falls from left to right.
For the given problem, the slope \( m = -1 \) tells us that the line descends steadily at a 45-degree angle. This indicates a balanced change between \( x \) and \( y \), so that for each one-unit increase in \( x \), \( y \) decreases by one unit. Understanding this helps you predict the behavior of the line on a graph just from looking at the equation.

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