finish the table

Question

y(1) = 1/2x + 5
y(2) = -2x -1

x y(1) = 1/2x + 5 y(2) = -2x -1 (x,y(1)) (x,y(2))
8
4
0
-2

Answer 1

Let's calculate the values of \( y(1) \) and \( y(2) \) for the given \( x \) values, and then write them in the requested table format.

Step-by-step calculations:

1. For \( x = 8 \):

   • \( y(1) = \frac{1}{2}(8) + 5 = 4 + 5 = 9 \)
   • \( y(2) = -2(8) - 1 = -16 - 1 = -17 \)
   • Points: \( (8, 9) \) and \( (8, -17) \)

2. For \( x = 4 \):

   • \( y(1) = \frac{1}{2}(4) + 5 = 2 + 5 = 7 \)
   • \( y(2) = -2(4) - 1 = -8 - 1 = -9 \)
   • Points: \( (4, 7) \) and \( (4, -9) \)

3. For \( x = 0 \):

   • \( y(1) = \frac{1}{2}(0) + 5 = 0 + 5 = 5 \)
   • \( y(2) = -2(0) - 1 = 0 - 1 = -1 \)
   • Points: \( (0, 5) \) and \( (0, -1) \)

4. For \( x = -2 \):

   • \( y(1) = \frac{1}{2}(-2) + 5 = -1 + 5 = 4 \)
   • \( y(2) = -2(-2) - 1 = 4 - 1 = 3 \)
   • Points: \( (-2, 4) \) and \( (-2, 3) \)

Compiled Table:

Now, we compile the results into a table format:

| \( x \) | \( y(1) = \frac{1}{2}x + 5 \) | \( y(2) = -2x - 1 \) | \( (x,y(1)) \) | \( (x,y(2)) \) | |---------|-----------------------------|-----------------------|----------------|----------------| | 8 | 9 | -17 | (8,9) | (8,-17) | | 4 | 7 | -9 | (4,7) | (4,-9) | | 0 | 5 | -1 | (0,5) | (0,-1) | | -2 | 4 | 3 | (-2,4) | (-2,3) |

This completes the table!

Answer 2

fill in the graph not table

Answer 3

Since I can't draw the graph directly here, I can guide you on how to plot the points from the equations \( y(1) = \frac{1}{2}x + 5 \) and \( y(2) = -2x - 1 \) on a graph.

Steps to Plot the Graph:

1. Set Up Your Axes:

   • Draw a horizontal line (x-axis) and a vertical line (y-axis).
   • Mark equal intervals on each axis.

2. Plot the Points for \( y(1) = \frac{1}{2}x + 5 \):

   • Calculate and plot the points for \( y(1) \):

     • For \( x = 8 \): \( y(1) = 9 \) → Point (8, 9)
     • For \( x = 4 \): \( y(1) = 7 \) → Point (4, 7)
     • For \( x = 0 \): \( y(1) = 5 \) → Point (0, 5)
     • For \( x = -2 \): \( y(1) = 4 \) → Point (-2, 4)

   • Connect these points with a straight line, as this function is linear.

3. Plot the Points for \( y(2) = -2x - 1 \):

   • Calculate and plot the points for \( y(2) \):

     • For \( x = 8 \): \( y(2) = -17 \) → Point (8, -17)
     • For \( x = 4 \): \( y(2) = -9 \) → Point (4, -9)
     • For \( x = 0 \): \( y(2) = -1 \) → Point (0, -1)
     • For \( x = -2 \): \( y(2) = 3 \) → Point (-2, 3)

   • Connect these points with a straight line, as this function is also linear.

Visual Representation:

Once you plot the points, you will have two lines on your graph:

• The line for \( y(1) \) will rise from the left to the right (positive slope).
• The line for \( y(2) \) will fall from the left to the right (negative slope).

Labeling:

• Label the first line as \( y(1) = \frac{1}{2}x + 5 \).
• Label the second line as \( y(2) = -2x - 1 \).

Example for Graphing (Sketch):

                     |
                  10 |            x
                     |
                   9 |x
                     |
                   8 |
                     |
                   7 |       x
                     |
                   6 |
                     |
                   5 |x
                     |
                   4 |                x
                     |   
                   3 |                      x
                     |
                   2 |
                     |
                   1 |                     
                     |________________________
                    -2 -1  0  1  2  3  4  5  6  7  8  9  10
                            x        

In this ASCII sketch, "x" represents the points you would plot for \( y(1) \) and \( y(2) \). Note that the actual graph will be a coordinate grid with much more detail.

Feel free to use graph paper or any graphing software to visualize this accurately!

Answer 4

fill in the missing spots not a actual graph

Answer 5

Got it! Let's fill in the values for \( y(1) \) and \( y(2) \) in the remaining spots of your previous inquiry, following the same format. Below is the complete list:

Given Equations:

• \( y(1) = \frac{1}{2}x + 5 \)
• \( y(2) = -2x - 1 \)

Values for Each \( x \):

| \( x \) | \( y(1) = \frac{1}{2}x + 5 \) | \( y(2) = -2x - 1 \) | \( (x,y(1)) \) | \( (x,y(2)) \) | |---------|-----------------------------|-----------------------|----------------|----------------| | 8 | \( 9 \) | \( -17 \) | \( (8, 9) \) | \( (8, -17) \) | | 4 | \( 7 \) | \( -9 \) | \( (4, 7) \) | \( (4, -9) \) | | 0 | \( 5 \) | \( -1 \) | \( (0, 5) \) | \( (0, -1) \) | | -2 | \( 4 \) | \( 3 \) | \( (-2, 4) \) | \( (-2, 3) \) |

Summary of the Results:

• For \( x = 8 \):

  • \( y(1) = 9 \)
  • \( y(2) = -17 \)

• For \( x = 4 \):

  • \( y(1) = 7 \)
  • \( y(2) = -9 \)

• For \( x = 0 \):

  • \( y(1) = 5 \)
  • \( y(2) = -1 \)

• For \( x = -2 \):

  • \( y(1) = 4 \)
  • \( y(2) = 3 \)

Feel free to use these filled values to visualize or graph the functions! If there's anything else you'd like, just let me know!

Answer 6

Graph the linear equations using a table of values.
y(1) =1/3x - 3
y(2) = -1/3x + 1

x
y(1)=
1/3x-3
y(2)=
-1/3x + 1
(x,y(1))
(x, y(2))
3

6

-9

Graph the equations to see the solution.

Answer 7

To graph the linear equations \( y(1) = \frac{1}{3}x - 3 \) and \( y(2) = -\frac{1}{3}x + 1 \), we can fill in the table of values for the specified \( x \) values and calculate the corresponding \( y \) values.

Step 1: Calculate \( y(1) \) and \( y(2) \) for each \( x \)

1. For \( x = 3 \):

   • \( y(1) = \frac{1}{3}(3) - 3 = 1 - 3 = -2 \)
   • \( y(2) = -\frac{1}{3}(3) + 1 = -1 + 1 = 0 \)

2. For \( x = 6 \):

   • \( y(1) = \frac{1}{3}(6) - 3 = 2 - 3 = -1 \)
   • \( y(2) = -\frac{1}{3}(6) + 1 = -2 + 1 = -1 \)

3. For \( x = -9 \):

   • \( y(1) = \frac{1}{3}(-9) - 3 = -3 - 3 = -6 \)
   • \( y(2) = -\frac{1}{3}(-9) + 1 = 3 + 1 = 4 \)

Summary Table of Values

Now, we can summarize these calculations in a table format:

| \( x \) | \( y(1) = \frac{1}{3}x - 3 \) | \( y(2) = -\frac{1}{3}x + 1 \) | \( (x, y(1)) \) | \( (x, y(2)) \) | |---------|-------------------------------|----------------------------------|-----------------|-----------------| | 3 | -2 | 0 | (3, -2) | (3, 0) | | 6 | -1 | -1 | (6, -1) | (6, -1) | | -9 | -6 | 4 | (-9, -6) | (-9, 4) |

Step 2: Graph the Equations

You can plot the points from this table onto a graph:

1. For \( y(1) = \frac{1}{3}x - 3 \):

   • Plot points: (3, -2), (6, -1), (-9, -6)

2. For \( y(2) = -\frac{1}{3}x + 1 \):

   • Plot points: (3, 0), (6, -1), (-9, 4)

Graphing Instructions:

• Draw the x-axis and y-axis, labeling them.
• Mark the points from the table on the graph.
• For each equation, connect the points with a straight line.

Expected Graph Behavior:

• The line \( y(1) \) has a positive slope and tilts upwards as \( x \) increases.
• The line \( y(2) \) has a negative slope and tilts downward as \( x \) increases.

Solution Intersection:

The solution to the system of equations occurs at the point where the two lines intersect. Based on the calculations, this point would be found by solving the equations concurrently or observing the graph.

Summary of Solutions:

Make sure to label your graph with the equations and highlight the intersection point, if you find it graphically. If you need any further assistance or clarification on any step, feel free to ask!