Problem 14 - Prealgebra

Question

For each equation, complete the given ordered pairs. $$y=-5 x \quad(0,),(-1,),(1,)$$

Step-by-Step Solution

Answer

(0, 0), (-1, 5), (1, -5)

Step 1: Understand the Equation

The given equation is a linear equation, where \( y = -5x \). This implies that for each value of \( x \), \( y \) is equal to \(-5\) times that value.

Step 2: Substitute x = 0

Substitute \( x = 0 \) into the equation and solve for \( y \): \[ y = -5(0) = 0 \]Hence, when \( x = 0 \), \( y = 0 \). The completed ordered pair is \((0, 0)\).

Step 3: Substitute x = -1

Substitute \( x = -1 \) into the equation and solve for \( y \): \[ y = -5(-1) = 5 \]Thus, when \( x = -1 \), \( y = 5 \). The completed ordered pair is \((-1, 5)\).

Step 4: Substitute x = 1

Substitute \( x = 1 \) into the equation and solve for \( y \): \[ y = -5(1) = -5 \]Therefore, when \( x = 1 \), \( y = -5 \). The completed ordered pair is \((1, -5)\).

Key Concepts

ordered pairs

Ordered pairs are fundamental in graphing and solving linear equations. They consist of two numbers: the first number represents the value on the x-axis, and the second one, on the y-axis. When working with equations, ordered pairs like

• \((0, 0)\)
• \((-1, 5)\)
• \((1, -5)\)

reflect solutions that satisfy the given equation.
In the equation \(y = -5x\), each solution represents a set of coordinates where the line intersects the coordinate plane. To find these pairs, you can substitute different values of \(x\) to determine \(y\). The final ordered pair provides a complete coordinate, showing us exactly where a point lies on the coordinate plane.

substitution method

The substitution method is a powerful tool for solving equations. It's particularly useful for finding specific values of unknown variables. Here's how it works in the context of the exercise provided:
To solve for \(y\) using the substitution method, you choose a value for \(x\) and substitute it into the equation. For example, if the equation is \(y = -5x\), and you choose \(x = 0\), you substitute it in and calculate:

• For \(x = 0\): \(y = -5(0) = 0\)
• For \(x = -1\): \(y = -5(-1) = 5\)
• For \(x = 1\): \(y = -5(1) = -5\)

This method enables you to fill in the placeholders in the ordered pairs with precise values. By repeating these steps for other values of \(x\), you can plot more points on the coordinate plane.

coordinate plane

The coordinate plane is a two-dimensional field where we graphically represent ordered pairs. It consists of two perpendicular lines: the horizontal x-axis and vertical y-axis. Each point on this plane corresponds to an ordered pair \((x, y)\) derived from your linear equation.
In the context of the equation \(y = -5x\), interpreting it graphically involves marking points like \((0, 0)\), \((-1, 5)\), and \((1, -5)\) on the coordinate plane.

• The x-value shows the position along the horizontal axis.
• The y-value indicates the position along the vertical axis.

Each ordered pair tells you exactly where to place a point on this plane, allowing you to visualize the line formed by the equation. By connecting these points, you get a good understanding of the graphical representation of linear relationships.