A linear equation is graphed on the coordinate plane below.

Question

The graph shows a coordinate grid with a straight line that passes through the points (negative 9, negative 5), (negative 6, negative 4) (negative 3, negative 3), (0, negative 2), (3, negative 1), (6, 0), and (9, 1).

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

Answer 1

To determine a second equation that would intersect with the given line at exactly one point, we need to create a line that does not have the same slope as the given line.

First, let’s find the slope of the line represented by the points you provided. We can choose any two points to calculate the slope, but let’s use (0, -2) and (3, -1):

1. The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points (0, -2) and (3, -1):

\[ m = \frac{-1 - (-2)}{3 - 0} = \frac{-1 + 2}{3} = \frac{1}{3} \]

So, the slope of the given line is \( \frac{1}{3} \).

To ensure that a new line intersects the original line at exactly one point, the new line must have a different slope. For example, we can choose a slope of \( -1 \).

Now, we can construct the equation of the new line using the point-slope form \( y - y_1 = m(x - x_1) \). Let’s use the point (0, 1) for simplicity with our chosen slope of \( -1 \):

\[ y - 1 = -1(x - 0) \]

This simplifies to:

\[ y = -x + 1 \]

Combining both lines:

• The given line (found earlier) is of the form \( y = \frac{1}{3}x - 2 \).
• The new line is \( y = -x + 1 \).

Both lines will cross at exactly one point, resulting in a system of equations with exactly one solution.

So the equation \( y = -x + 1 \) is appropriate for your requirement.

Answer 2

wrong

Answer 3

I apologize for the error. Let’s carefully reanalyze the problem and find a suitable equation that intersects the given line at exactly one point.

Step 1: Determine the slope and equation of the given line.

Using the points you provided, let’s ensure we find the correct equation of the line that passes through the given points.

We already calculated the slope \( m \) using (0, -2) and (3, -1):

\[ m = \frac{-1 - (-2)}{3 - 0} = \frac{1}{3} \]

Next, we can use the point-slope form of a line equation \( y - y_1 = m(x - x_1) \) with point (0, -2):

\[ y - (-2) = \frac{1}{3}(x - 0) \] \[ y + 2 = \frac{1}{3}x \] \[ y = \frac{1}{3}x - 2 \]

Step 2: Create a new line with a different slope.

To have a system of equations with exactly one solution, we need a new line with a different slope from \( \frac{1}{3} \). Let’s choose a slope of, for example, \( 2 \).

Step 3: Write the equation.

Using point (0, 1) on the y-axis to ensure it has the correct y-intercept, the equation would be:

Using slope-intercept form: \[ y = mx + b \] Here, we have \( m = 2 \) and \( b = 1\).

So the equation of the new line is: \[ y = 2x + 1 \]

Conclusion

The equation \( y = 2x + 1 \) should intersect the original line \( y = \frac{1}{3}x - 2 \) at exactly one point, thus creating a system of equations with exactly one solution.

Thank you for your patience, and I hope this correction helps!