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 create a system of equations with exactly one solution, we need to introduce a line that intersects the given line at exactly one point.

First, let's determine the slope and y-intercept of the line described by the points given in your question:

1. The points (0, -2) and (3, -1) can be used to find the slope: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-2)}{3 - 0} = \frac{1}{3} \]

2. Using the slope of \(\frac{1}{3}\) and the point (0, -2), we can use the slope-intercept form of the line, \(y = mx + b\): \[ y = \frac{1}{3}x - 2 \]

Next, to find a second line that intersects this line at exactly one point, we can choose a different slope that is not equal to \(\frac{1}{3}\) (the slope of the original line). Let's say we choose a slope of 2, for example.

We can also choose any y-intercept that’s not equal to -2 (the y-intercept of the original line) to ensure they intersect. A possible equation could be: \[ y = 2x + 2 \]

Now, the system of equations we have is:

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

Because these two lines have different slopes, they will intersect at exactly one point, resulting in a system of equations with exactly one solution.

Thus, the equation that results in one solution is: \[ y = 2x + 2 \]