Question

A graph is shown
Point 1: (0, 6)
Point 2: (3, -1)
Point 3: (0, -2)
Point 4: (-3, -3)

When graphed on the same coordinate plane, which equation results in a system of equations with exactly one solution?

A. -2x + 6y = -12

B. 3x - 9y = 9

C. x - 3y = 6

D. 9x - 3y = 6

Answer 1

To determine which of the given equations will intersect the lines formed by the points provided (Point 1: (0, 6), Point 2: (3, -1), Point 3: (0, -2), Point 4: (-3, -3)) at exactly one point, we first need to calculate the slopes of the lines formed by these points, and then choose an equation that does not have parallel lines.

Let's determine the equations of the lines formed by the given points:

**Between Point 1 (0, 6) and Point 2 (3, -1):**
- Slope (m) = (y2 - y1)/(x2 - x1) = (-1 - 6) / (3 - 0) = -7/3
- Using Point 1 to find the equation:
\( y - 6 = -\frac{7}{3}(x - 0) \)
\( y = -\frac{7}{3}x + 6 \)

**Between Point 3 (0, -2) and Point 4 (-3, -3):**
- Slope (m) = (-3 - (-2)) / (-3 - 0) = -1 / -3 = 1/3
- Using Point 3 to find the equation:
\( y + 2 = \frac{1}{3}(x - 0) \)
\( y = \frac{1}{3}x - 2 \)

Now that we have two equations of lines:
1. \( y = -\frac{7}{3}x + 6 \)
2. \( y = \frac{1}{3}x - 2 \)

For a system of equations to have exactly one solution, the lines must intersect at a single point, which means they must not be parallel.

Now let's convert the provided options into slope-intercept form (y = mx + b) to check their slopes:

**A. \( -2x + 6y = -12 \)**
Rearranging gives:
\( 6y = 2x - 12 \) or \( y = \frac{1}{3}x - 2 \)

This has a slope of 1/3, which is the same as line 2 and is therefore parallel.

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**B. \( 3x - 9y = 9 \)**
Rearranging gives:
\( -9y = -3x + 9 \) or \( y = \frac{1}{3}x - 1 \)

This has a slope of 1/3, which is the same as line 2 and is therefore parallel.

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**C. \( x - 3y = 6 \)**
Rearranging gives:
\( -3y = -x + 6 \) or \( y = \frac{1}{3}x - 2 \)

This also has a slope of 1/3, which is the same as line 2 and is therefore parallel.

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**D. \( 9x - 3y = 6 \)**
Rearranging gives:
\( -3y = -9x + 6 \) or \( y = 3x - 2 \)

This has a slope of 3, which is different from both previous slopes.

Since line D is not parallel to either of the two lines we derived from the points, it will intersect both lines at different points, giving the system exactly one solution.

Thus, the correct answer is:
**D. 9x - 3y = 6**