What is the standard form of the following equation?

Question

y + 3 = 1/4x
Use integers for A, B, and C. Enter your answer in the box.

Answer 1

To convert the given equation \( y + 3 = \frac{1}{4}x \) into standard form, we need to rearrange it to the form \( Ax + By = C \), where \( A\), \( B\), and \( C\) are integers.

1. Start with the original equation: \[ y + 3 = \frac{1}{4}x \]

2. Subtract 3 from both sides: \[ y = \frac{1}{4}x - 3 \]

3. To eliminate the fraction, multiply every term by 4: \[ 4y = x - 12 \]

4. Rearrange the equation to get all terms on one side: \[ -x + 4y = -12 \]

5. To standardize the format, multiply the entire equation by -1 to make \( A \) positive: \[ x - 4y = 12 \]

Now, the equation is in standard form \( Ax + By = C\) with \( A = 1\), \( B = -4\), and \( C = 12\).

Thus, the standard form of the equation is: \[ \boxed{x - 4y = 12} \]

Answer 2

Graph x and y.
4x + 2y = 6

Answer 3

To graph the equation \(4x + 2y = 6\), we can follow these steps:

1. Rewrite the equation in slope-intercept form (y = mx + b): \[ 2y = -4x + 6 \] \[ y = -2x + 3 \] Here, the slope \(m\) is -2, and the y-intercept \(b\) is 3.

2. Plot the y-intercept: Start by plotting the point where the line intersects the y-axis. Since the y-intercept is 3, the point is (0, 3).

3. Use the slope to find another point: The slope -2 means that for every 1 unit you move to the right (positive x-direction), you move 2 units down (negative y-direction).

   • From the point (0, 3), moving 1 unit to the right to x = 1 gives: \[ y = -2(1) + 3 = 1 \]
   • So another point is (1, 1).

4. Plot another point using the slope in the opposite direction:

   • From (0, 3) moving 1 unit to the left (negative x-direction) gives: \[ y = -2(-1) + 3 = 5 \]
   • So the point is (-1, 5).

5. Draw the line: Now that you have the points (0, 3), (1, 1), and (-1, 5), you can draw a straight line through these points.

Summary of Points

• The y-intercept at (0, 3).
• Another point at (1, 1).
• Another point at (-1, 5).

Final Graph

To summarize:

• Mark the points on a coordinate grid: (0, 3), (1, 1), and (-1, 5).
• Draw a straight line through these points. This line represents the equation \(4x + 2y = 6\).

If you would like a visual representation, you can use graphing software or a graphing calculator. However, the steps above will allow you to sketch it on paper.

Answer 4

Graph x and y.
y = −1/3x + 4

Answer 5

To graph the equation \(y = -\frac{1}{3}x + 4\), follow these steps:

Step 1: Identify the slope and y-intercept

The equation is already in slope-intercept form \(y = mx + b\), where:

• The slope \(m = -\frac{1}{3}\).
• The y-intercept \(b = 4\).

This means the line crosses the y-axis at the point (0, 4).

Step 2: Plot the y-intercept

Start by plotting the y-intercept on the graph:

• Point: (0, 4)

Step 3: Use the slope to find another point

The slope of \(-\frac{1}{3}\) means:

• For every 3 units you move to the right (positive direction on the x-axis), you will move down 1 unit (negative direction on the y-axis).

From the point (0, 4):

• Move 3 units to the right to \(x = 3\): \[ y = -\frac{1}{3}(3) + 4 = -1 + 4 = 3 \]
• So another point is (3, 3).

Step 4: Plot the second point

Now, plot the second point:

• Point: (3, 3)

Step 5: Find a third point (optional)

You can also find a third point using the slope in the opposite direction:

• From (0, 4), move 3 units to the left (to \(x = -3\)): \[ y = -\frac{1}{3}(-3) + 4 = 1 + 4 = 5 \]
• So another point is (-3, 5).

Step 6: Plot the third point

Now, plot the third point:

• Point: (-3, 5)

Step 7: Draw the line

Now that you have the points:

• (0, 4)
• (3, 3)
• (-3, 5)

Connect these points with a straight line extending in both directions. This line represents the equation \(y = -\frac{1}{3}x + 4\).

Summary of Points

• Y-intercept: (0, 4)
• Another point: (3, 3)
• Optional point: (-3, 5)

Final Notes

• The line will slope downward from left to right due to the negative slope.
• You can use graphing software or a graphing calculator to get a visual representation, but sketching it on a grid with the points you've plotted will also work well.

Answer 6

What are the plots?

Answer 7

To provide a visual representation of the graph for the equation \(y = -\frac{1}{3}x + 4\), here are the steps to plot it, along with a text description of what it would look like:

Points to Plot

1. Y-intercept (0, 4): This point is where the line crosses the y-axis. It's plotted above the origin on the y-axis.
2. Point (3, 3): Move 3 units to the right from the origin (0, 0) and then down 1 unit to reach this point.
3. Optional Point (-3, 5): Move 3 units to the left of the origin and then up 1 unit to reach this point.

Plotting the Points

1. Draw the axes: A vertical line for the y-axis and a horizontal line for the x-axis.
2. Plot the points:
   • (0, 4): This point is plotted on the y-axis at 4.
   • (3, 3): This point is located 3 units to the right and 3 units down from the y-axis.
   • (-3, 5): This point is located 3 units to the left and 5 units up from the x-axis.

Drawing the Line

• After plotting the points, draw a straight line through them, extending the line in both directions.

Textual Description of the Graph

• The graph will slope downwards from left to right, showing that as \(x\) increases, \(y\) decreases.
• The line will intersect the y-axis at the point (0, 4) and will gradually decrease as it moves towards the right.

Example Graph Representation

Since I cannot create visual graphics directly, here's a simple example of how to visualize it with text:

Y-axis
 |
5 |       ( -3, 5 )
4 |  ( 0, 4 )
3 |              ( 3, 3 )
2 |
1 |
0 +--------------------X-axis
  -3  -2  -1  0  1  2  3

In an actual graph, the line would smoothly connect the points (0, 4), (3, 3), and (-3, 5), extending infinitely in both directions.

If You Need a Graph

You can use graphing tools or software like Desmos, GeoGebra, or even graphing calculators to plot this equation visually and see the line's precise behavior. If you have graph paper, you can manually plot the points and draw the line as described.