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Magazine Chapter 3: Problem 45
Construct a table of solutions and then graph equation. \(y=-6 x\)
Short Answer
Expert verified
Create a table with x and y values, then plot these points and connect them to graph the line y = -6x.
Step by step solution
01
Choose Values for x
To construct a table of solutions, start by selecting a range of values for \(x\). For simplicity, choose values such as \(-2, -1, 0, 1,\) and \(2\).
02
Calculate Corresponding y Values
Substitute each chosen \(x\) value into the equation \(y = -6x\) to find the corresponding \(y\) value.\(-\) If \(x = -2, y = -6(-2) = 12\).\(-\) If \(x = -1, y = -6(-1) = 6\).\(-\) If \(x = 0, y = -6(0) = 0\).\(-\) If \(x = 1, y = -6(1) = -6\).\(-\) If \(x = 2, y = -6(2) = -12\).
03
Create the Table of Solutions
Now, create a table with the \(x\) values and their corresponding \(y\) values.\(\begin{array}{c|c}x & y \ \hline-2 & 12 \-1 & 6 \0 & 0 \1 & -6 \2 & -12 \ \end{array}\)
04
Graph the Equation
Plot the \(x\) and \(y\) pairs from the table on a coordinate plane. The points are \((-2, 12), (-1, 6), (0, 0), (1, -6), (2, -12)\). Connect these points with a straight line to graph the equation \(y = -6x\). This line extends infinitely in both directions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solution Table
Understanding the construction of a solution table is an essential part of solving linear equations by graphing. When you work with a linear equation like \( y = -6x \), the objective is to find various solutions that satisfy the equation.
This is done by selecting simple values for \( x \) and calculating their corresponding \( y \) values.
Here are the steps broken down:
This is done by selecting simple values for \( x \) and calculating their corresponding \( y \) values.
Here are the steps broken down:
- Select Values: Choose a range of simple \( x \) values. In our example, we choose \(-2, -1, 0, 1,\) and \(2\). These values are easy to compute and provide a broad range to visualize the equation.
- Calculate \( y \) Values: Insert each \( x \) value into your equation to solve for \( y \). For instance, \( y = -6(-2) = 12 \) for \( x = -2 \). This process is repeated for each chosen \( x \) value to get a full set of \( x \) and \( y \) pairs.
- Create a Table: With the \( x \) and \( y \) pairs computed, organize them into a table. This table will help systematically show all possible solutions you have calculated.
Graphing Equations
Once your solution table is ready, the next step is to graph the equation, which in this case is \( y = -6x \).
Graphing a linear equation helps you visualize the relationship between \( x \) and \( y \) values provided by your solution table. Here's how you can accomplish this:
The graph demonstrates that for every decrease in \( x \), \( y \) decreases consistently at a rate determined by the slope, which in this case is -6, indicating that \( y \) decreases by 6 units for every unit \( x \) increases.
Graphing a linear equation helps you visualize the relationship between \( x \) and \( y \) values provided by your solution table. Here's how you can accomplish this:
- Plot the Points: Transfer the \( x, y \) pairs from your solution table onto the graph. For instance, start with the pair \( (-2, 12) \) and mark this point on the graph. Repeat this step for all pairs: \( (-1, 6), (0, 0), (1, -6), and (2, -12) \).
- Draw the Line: Connect the plotted points with a straight line. Since it's a linear equation, the plotted points should lie on a straight line if calculated correctly.
The graph demonstrates that for every decrease in \( x \), \( y \) decreases consistently at a rate determined by the slope, which in this case is -6, indicating that \( y \) decreases by 6 units for every unit \( x \) increases.
Coordinate Plane
The coordinate plane is the backdrop upon which all points from your solution table are plotted.
This is a two-dimensional plane defined by the \( x \)-axis and the \( y \)-axis, intersecting perpendicularly and forming four quadrants. Here's how we use and interpret the coordinate plane:
This is a two-dimensional plane defined by the \( x \)-axis and the \( y \)-axis, intersecting perpendicularly and forming four quadrants. Here's how we use and interpret the coordinate plane:
- The Axes: The horizontal line is called the \( x \)-axis, while the vertical line is the \( y \)-axis. The point where they cross is the origin \((0,0)\).
- Quadrants: The plane is divided into four sections called quadrants. The top-right section is Quadrant I, top-left is Quadrant II, bottom-left is Quadrant III, and bottom-right is Quadrant IV. Depending on the sign (positive or negative) of \( x \) and \( y \), the plotted point will fall into one of these quadrants.
- Plotting: To plot a point like \((-2, 12)\), start at the origin. Move left two units along the \( x \)-axis (since it's -2), and then move up twelve units (since it's 12 on the \( y \)-axis).
