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Use the image to answer the question.

A coordinate plane shows an x-axis ranging from 0 to 10 in increments of 2 and a y-axis ranging from 0 to 15 in increments of 3. The plotted points form an upward diagonal line. The first data point is at left parenthesis 2 comma 3 right parenthesis. The fourth data point is at left parenthesis 8 comma 12 right parenthesis. Two points in between the given points are left blank with only the parenthesis. The X-axis value of the second data points are the scale value next to 2 and the Y-axis value next to 3. The X-axis value of the third data points are the scale value next to 4 and the Y-axis value next to 6.

Choose the coordinate pair to plot on the coordinate plane so the ratio stays equivalent.

(1 point)
Responses

(4, 10)
Left Parenthesis 4 comma 10 Right Parenthesis

(3, 8)
Left Parenthesis 3 comma 8 Right Parenthesis

(4, 6)
Left Parenthesis 4 comma 8 Right Parenthesis

(6, 4)

11 answers

To maintain the equivalent ratio along the line that is plotted on the coordinate plane, we need to determine the pattern established by the given points.

The two given points are:

  1. (2, 3)
  2. (8, 12)

Let's find the slope between these two points: The change in \(y\) is \(12 - 3 = 9\) and the change in \(x\) is \(8 - 2 = 6\). Thus, the slope is \(\frac{9}{6} = \frac{3}{2}\).

This means for every 2 units change in \(x\), \(y\) changes by 3 units.

Using the established pattern, we can calculate additional points:

  • Starting from (2, 3) and moving to the next point, if we increase \(x\) by 2 (to x=4), we should increase \(y\) by 3 (to y=6). This gives us (4, 6).
  • If we increase \(x\) by another 2 (to x=6), we increase \(y\) by 3 again (to y=9), which gives us (6, 9).
  • Finally, from (6, 9), another increase of 2 in \(x\) (to x=8) results in \(y\) increasing by another 3 (to y=12), which we've already established.

Now we can check the answer choices:

  • (4, 10): This does not maintain the ratio since \(y\) is 10 while \(x\) is 4.
  • (3, 8): This does not maintain the ratio as well.
  • (4, 8): This does not maintain the ratio since the ratio should be \(6\) for 4.
  • (6, 4): This does not keep the ratio.

Based on the established pattern, the correct coordinate pair to maintain the ratio is (4, 6).

Thus, the answer is: Left Parenthesis 4, 6 Right Parenthesis

so, (6,4) ?

No, (6, 4) does not maintain the ratio established by the points (2, 3) and (8, 12). The slope is consistent with the ratio of \( \frac{y}{x} = \frac{3}{2}\), meaning for every 2 units increase in \(x\), \(y\) increases by 3.

Using this pattern:

  • For \(x = 2\), \(y = 3\) (which can be simplified to \(\frac{3}{2}\))
  • Next, for \(x = 4\), \(y\) should be \(6\) (which is \( \frac{6}{4} = \frac{3}{2}\))
  • Continuing, for \(x = 6\), \(y\) would be \(9\)
  • Finally, for \(x = 8\), \(y = 12\) (again, this follows the slope)

Therefore, the correct pattern produces (4, 6) as a corresponding point—a ratio of 3 to 2—rather than (6, 4) which introduces an incorrect ratio.

So the answer to keep the ratio equivalent along the line would still be: Left Parenthesis 4, 6 Right Parenthesis

Not (6, 4).

So which one:
1. (4, 10)
Left Parenthesis 4 comma 10 Right Parenthesis

2. (3, 8)
Left Parenthesis 3 comma 8 Right Parenthesis

3. (4, 6)
Left Parenthesis 4 comma 8 Right Parenthesis
or
4. (6, 4)

To maintain the ratio established by the points (2, 3) and (8, 12), we need to follow the slope derived from those points. Based on our previous discussion, the point (4, 6) maintains the ratio of \( \frac{3}{2} \), which is consistent with the established pattern.

Thus, from the provided options, the correct choice is:

3. (4, 6)

  • Left Parenthesis 4 comma 6 Right Parenthesis

The other options do not maintain the same ratio.

Use the image to answer the question.

A coordinate plane shows red paint on the X-axis and blue paint on the Y-axis. The X-axis ranges from 0 to 10 in increments of 2. The Y-axis ranges from 0 to 15 in increments of 3. The data points are left parenthesis 2 comma 3 right parenthesis, left parenthesis 4 comma 6 right parenthesis, left parenthesis 6 comma 9 right parenthesis, and left parenthesis 8 comma 12 right parenthesis.

Choose the coordinate pair to plot on the coordinate plane so the ratio stays equivalent.

(1 point)
Responses

(15, 10)

(10, 15)

(9, 15)

(10, 14)

To maintain the ratio established by the points given in the coordinate plane, we first need to examine the existing points:

  • (2, 3)
  • (4, 6)
  • (6, 9)
  • (8, 12)

From the points, we can see that the ratio of \(y\) to \(x\) is consistently \(\frac{3}{2}\):

  • For (2, 3), \(\frac{3}{2} = 1.5\)
  • For (4, 6), \(\frac{6}{4} = 1.5\)
  • For (6, 9), \(\frac{9}{6} = 1.5\)
  • For (8, 12), \(\frac{12}{8} = 1.5\)

To find a new point \((x, y)\) that maintains the same ratio, we want \( \frac{y}{x} = \frac{3}{2} \).

Rearranging that, we can express \(y\) in terms of \(x\): \[ y = \frac{3}{2}x \]

Now let's analyze each of the options provided:

  1. (15, 10): \[ \frac{10}{15} = \frac{2}{3} \] (not equivalent)

  2. (10, 15): \[ \frac{15}{10} = 1.5 \] (equivalent)

  3. (9, 15): \[ \frac{15}{9} \approx 1.67 \] (not equivalent)

  4. (10, 14): \[ \frac{14}{10} = 1.4 \] (not equivalent)

Based on this analysis, the only point that maintains the equivalent ratio of \(\frac{3}{2}\) is:

(10, 15)

Use the image to answer the question.

A coordinate plane shows red paint on the X-axis and blue paint on the Y-axis. The X-axis ranges from 0 to 10 in increments of 2. The Y-axis ranges from 0 to 15 in increments of 3. The data points are left parenthesis 2 comma 3 right parenthesis, left parenthesis 4 comma 6 right parenthesis, left parenthesis 6 comma 9 right parenthesis, and left parenthesis 8 comma 12 right parenthesis.

What is the ratio of red paint to blue paint?

(1 point)
Responses

3:4
ratio of 3 to 4

2:3
ratio of 2 to 3

1:2
ratio of 1 to 2

1:3

To determine the ratio of red paint to blue paint, we look at the relationship between the x-values (representing red paint) and the y-values (representing blue paint) from the given data points:

The data points are:

  1. (2, 3)
  2. (4, 6)
  3. (6, 9)
  4. (8, 12)

The ratios of \(y\) (blue paint) to \(x\) (red paint) can be observed in these points:

For each point:

  • For (2, 3), the ratio is \( \frac{3}{2} \) or 3:2 (blue:red)
  • For (4, 6), the ratio is \( \frac{6}{4} \) or 3:2 (blue:red)
  • For (6, 9), the ratio is \( \frac{9}{6} \) or 3:2 (blue:red)
  • For (8, 12), the ratio is \( \frac{12}{8} \) or 3:2 (blue:red)

This means that the amount of blue paint compared to red paint is consistent. To find the ratio of red paint to blue paint, we need to flip the ratio \(3:2\) to get the red paint in the numerator:

The corresponding ratio of red paint to blue paint is:

2:3

So, the answer is: 2:3 ratio of 2 to 3.

Use the image to answer the question.

A coordinate plane shows an x-axis ranging from 0 to 5 in increments of 1 and a y-axis ranging from 0 to 20 in increments of 4. The data points are left parenthesis 2 comma 8 right parenthesis, left parenthesis 3 comma 12 right parenthesis, left parenthesis 4 comma 16 right parenthesis, and left parenthesis 5 comma 20 right parenthesis.

Choose the coordinate pair to plot on the coordinate plane so the ratio stays equivalent.

(1 point)
Responses

(1, 3)
Left Parenthesis 1 comma 3 Right Parenthesis

(0, 4)
Left Parenthesis 0 comma 4 Right Parenthesis

(4, 1)
Left Parenthesis 4 comma 1 Right Parenthesis

(1, 4)
Left Parenthesis 1 comma 4 Right Parenthesis
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To determine which coordinate pair to choose so that the ratio stays equivalent, we first need to analyze the given points:

The data points are:

  1. (2, 8)
  2. (3, 12)
  3. (4, 16)
  4. (5, 20)

From these points, we can see a pattern:

The ratios of \(y\) to \(x\) can be calculated as follows:

  • For (2, 8): \(\frac{y}{x} = \frac{8}{2} = 4\)

  • For (3, 12): \(\frac{y}{x} = \frac{12}{3} = 4\)

  • For (4, 16): \(\frac{y}{x} = \frac{16}{4} = 4\)

  • For (5, 20): \(\frac{y}{x} = \frac{20}{5} = 4\)

Now, we observe that the ratio \(y:x\) is consistently \(4:1\).

To maintain this ratio of \(4:1\) (or \(4\) units of \(y\) for every \(1\) unit of \(x\)), we can express that as:

\[ y = 4x \]

Now, let's check the provided coordinate pairs to find which one maintains this ratio:

  1. (1, 3): \[ \frac{3}{1} = 3 \] (not equivalent)

  2. (0, 4): \[ \frac{4}{0} \] (undefined, not valid)

  3. (4, 1): \[ \frac{1}{4} = 0.25 \] (not equivalent)

  4. (1, 4): \[ \frac{4}{1} = 4 \] (equivalent)

Based on this evaluation, the only coordinate pair that maintains the same ratio of \(y:x = 4:1\) is:

(1, 4)

  • Left Parenthesis 1 comma 4 Right Parenthesis

This choice keeps the equivalent ratio consistent with the established points on the coordinate plane.