Sure, let's go through the process of plotting the linear relationship [tex]\( y = 2x - 1 \)[/tex] on a Cartesian plane, using a table of values for clarity.
### Step-by-Step Solution:
1. Generate Values for x:
Create a list of [tex]\( x \)[/tex]-values. Let's use the following values for [tex]\( x \)[/tex]:
[tex]\[ x = -2, -1, 0, 1, 2 \][/tex]
2. Calculate Corresponding y-values:
Use the equation [tex]\( y = 2x - 1 \)[/tex] to calculate the corresponding [tex]\( y \)[/tex]-values for each [tex]\( x \)[/tex].
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 2(-2) - 1 = -4 - 1 = -5 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 2(-1) - 1 = -2 - 1 = -3 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) - 1 = 0 - 1 = -1 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 2(1) - 1 = 2 - 1 = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2(2) - 1 = 4 - 1 = 3 \][/tex]
3. Complete the Table of Values:
Fill in the [tex]\( y \)[/tex]-values in the table based on the calculations above.
[tex]\[
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$y$ & -5 & -3 & -1 & 1 & 3 \\
\hline
\end{tabular}
\][/tex]
4. Plot on the Cartesian Plane:
Use the [tex]\((x, y)\)[/tex] pairs from the table to plot the points on a Cartesian plane. After plotting the points, draw a straight line through them, since [tex]\( y = 2x - 1 \)[/tex] is a linear relationship.
The points to plot are:
- [tex]\((-2, -5)\)[/tex]
- [tex]\((-1, -3)\)[/tex]
- [tex]\((0, -1)\)[/tex]
- [tex]\((1, 1)\)[/tex]
- [tex]\((2, 3)\)[/tex]
5. Drawing the Line:
- Start by plotting point [tex]\((-2, -5)\)[/tex].
- Then plot [tex]\((\-1, -3)\)[/tex], [tex]\((0, -1)\)[/tex], [tex]\((1, 1)\)[/tex], [tex]\((2, 3)\)[/tex] sequentially.
- Draw a straight line through all the plotted points.
When you draw the Cartesian plane and plot these points, you will see that they form a straight line, which visually represents the equation [tex]\( y = 2x - 1 \)[/tex].
### Summary:
The linear relationship [tex]\( y = 2x - 1 \)[/tex] is plotted on the Cartesian plane by calculating [tex]\( y \)[/tex]-values for chosen [tex]\( x \)[/tex]-values and plotting the resulting coordinate points. The table of values helps organize and visualize the points, and connecting these points with a straight line represents the equation graphically.
