Problem 8 - Prealgebra

Question

For each equation, complete the given ordered pairs. $$y=3 x-5 \quad(, 13),(0,),(-2,)$$

Step-by-Step Solution

Answer

Completed ordered pairs: (6, 13), (0, -5), (-2, -11).

Step 1: Solve for the missing x-value in the pair (?, 13)

To find the missing x-value, set y to 13 and plug into the equation: \[ 13 = 3x - 5 \]Add 5 to both sides:\[ 18 = 3x \]Divide both sides by 3:\[ x = 6 \]So the completed ordered pair is (6, 13).

Step 2: Solve for y when x = 0 in the pair (0, ?)

Plug x = 0 into the equation:\[ y = 3(0) - 5 \]Simplify the equation:\[ y = -5 \]Thus, the completed ordered pair is (0, -5).

Step 3: Solve for y when x = -2 in the pair (-2, ?)

Plug x = -2 into the equation:\[ y = 3(-2) - 5 \]Calculate the expression:\[ y = -6 - 5 \]\[ y = -11 \]Thus, the completed ordered pair is (-2, -11).

Key Concepts

Ordered Pairs

Ordered pairs are essential concepts in mathematics, especially when working on linear equations. An ordered pair comprises two elements, typically written in the form \((x, y)\). This represents a point on the coordinate plane where:

• The first element, \(x\), indicates the horizontal position.
• The second element, \(y\), signifies the vertical position.

Each ordered pair gives precise information about a location in a two-dimensional space. Understanding how to determine or "complete" ordered pairs involves knowing how to manipulate algebraic expressions to find unknown \(x\) or \(y\) values.
For example, in completing an ordered pair like \((?, 13)\), you need to determine what \(x\) corresponds to \(y\) when \(y\) equals 13 in the given equation. Solve the equation by plugging in the known \(y\) and solving for \(x\), just as demonstrated in the step-by-step solution.

Graphing Equations

Graphing equations is a vital skill as it visually represents algebraic relationships. A linear equation like \(y = 3x - 5\) represents a straight line on a graph.
Here's how graphing aids understanding:

• It transforms abstract algebraic numbers into sensible, visual lines.
• You can quickly see the relationship between \(x\) and \(y\). If \(x\) increases, \(y\) changes according to the equation.

To graph such equations, it's essential first to determine multiple ordered pairs. Use these pairs to find points on the graph. Once plotted, connect the dots to visualize the line. This process not only verifies the algebraic work but also gains insights into the function's behavior, such as its slope, intercept, and the direction it follows across the graph.

Coordinate Plane

A coordinate plane, also known as the Cartesian plane, is a flat surface divided into four quadrants by a horizontal line and a vertical line.
Here are its key features:

• The horizontal line is called the x-axis, while the vertical line is the y-axis.
• These axes intersect at the "origin," denoted as \((0, 0)\).
• Each point on this plane is determined by an ordered pair \((x, y)\).

The coordinate plane is crucial when graphing equations as it provides the space where these graphs can be drawn. By plotting ordered pairs onto this plane, one can visualize solutions and understand the spatial arrangement and movement patterns of mathematical equations. Effective use of a coordinate plane transforms theoretical mathematical concepts into tangible forms, making it easier to analyze and interpret linear relationships.