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Chapter 8: Problem 20

Use the rectangular coordinate system below each exercise to plot the three ordered pair solutions of the given equation. $$ 3 x+y=5 ;(1,2),(0,5),(2,-1) $$

Short Answer

Expert verified
Plot points (1,2), (0,5), and (2,-1) on a graph to form a line.

Step by step solution

01

Understand the Equation

The equation given is \(3x + y = 5\). This is a linear equation in two variables, where each ordered pair \((x, y)\) should satisfy the equation when plugged into it.
02

Verify Each Ordered Pair

We must check if each ordered pair \((x, y)\) satisfies the equation. - For \((1, 2)\): Substitute \(x = 1\) and \(y = 2\). We have \(3(1) + 2 = 3 + 2 = 5\). - For \((0, 5)\): Substitute \(x = 0\) and \(y = 5\). We have \(3(0) + 5 = 0 + 5 = 5\).- For \((2, -1)\): Substitute \(x = 2\) and \(y = -1\). We have \(3(2) + (-1) = 6 - 1 = 5\). All pairs satisfy the equation.
03

Plot the Ordered Pairs on the Graph

Using the rectangular coordinate system, plot each of the three pairs on the graph:- Plot \((1, 2)\): Locate \(x = 1\) on the horizontal axis and \(y = 2\) on the vertical axis, then place a point there.- Plot \((0, 5)\): Locate \(x = 0\) on the horizontal axis and \(y = 5\) on the vertical axis, then place a point there.- Plot \((2, -1)\): Locate \(x = 2\) on the horizontal axis and \(y = -1\) on the vertical axis, then place a point there.
04

Draw the Line

Since all plotted points satisfy the equation \(3x + y = 5\), they should form a straight line. Connect these points by drawing a line through them. This line represents all solutions of the equation.

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Key Concepts

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

Coordinate System
The coordinate system is a fundamental concept in mathematics and crucial for graphing equations. Imagine a plane with two perpendicular lines crossing at a point called the origin. These lines form the axes of the coordinate system: the horizontal line called the x-axis, and the vertical line named the y-axis. Together, they help you pinpoint exact locations in 2D space.

Each point on this plane is described using a pair of numbers, known as an ordered pair, which we will explore next. Understanding how to navigate this grid will boost your ability to visualize and interpret mathematical relationships effectively.
Ordered Pairs
An ordered pair is a simple yet powerful concept that tells you exactly where a point lies on the coordinate plane. It is written as \((x, y)\), where the first number corresponds to a position on the x-axis and the second to the y-axis. Think of the ordered pair \((x, y)\) as directions to a treasure buried on a map!

For example, the ordered pair \((1, 2)\) means you move 1 unit right along the x-axis and then 2 units up along the y-axis. This allows each point on the plane to have its own unique address, helping you identify the position and relationship of multiple points, especially when you're plotting them on a graph.
Graph Plotting
Graph plotting is a visual way of representing equations, showing the relationships between different sets of values. To plot points, you first identify the ordered pair that represents their position. Then, you locate these coordinates on the coordinate plane and mark the spots.

For the linear equation \(3x + y = 5\), you'll plot ordered pairs like \((1, 2)\), \((0, 5)\), and \((2, -1)\). This involves starting at the origin, moving along the x-axis, and then along the y-axis to find the point:
  • For \((1, 2)\), move 1 unit right and 2 units up.
  • For \((0, 5)\), stay at the y-axis and move 5 units up.
  • For \((2, -1)\), move 2 units right and 1 unit down.
Plotting these points and connecting them forms a straight line, illustrating all solutions to the equation.
Solutions Verification
Solutions verification is a crucial step to ensure your calculations are correct. When working with linear equations like \(3x + y = 5\), you need to confirm that each point (ordered pair) is a solution to the equation.

Verification involves substituting the x and y values from each ordered pair into the equation:
  • For \((1, 2)\), replacing gives \(3(1) + 2 = 5\), which is true.
  • For \((0, 5)\), replacing gives \(3(0) + 5 = 5\), also true.
  • For \((2, -1)\), replacing yields \(3(2) - 1 = 5\), again confirming it as a solution.
Each true statement confirms that the ordered pair is indeed a solution of the linear equation, ensuring accuracy before you move ahead with plotting these on a graph.

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Most popular questions from this chapter

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Solve. In your own words, explain why the probability of an event cannot be greater than \(1 .\)
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