Question

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

Responses

2x−y=1
2 x − y = 1

4x−2y=−4
4 x − 2 y = − 4

4x−2y=−2
4 x − 2 y = − 2

2x−3y=6​

Answer 1

A system of equations with exactly one solution, we can analyze the slopes of the equations. If the slopes are different, the system will have one unique solution.

If the slopes are different, the system will have one unique solution. Let's write the equations in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:

1. 2x - y = 12 can be rewritten as y = 2x - 12. The slope is 2.
2. 4x - 2y = -4 can be simplified to 2x - y = -2, which can be further simplified to y = 2x + 2. The slope is 2.
3. 4x - 2y = -2 can be simplified to 2x - y = -1, which can be further simplified to y = 2x + 1. The slope is 2.
4. 2x - 3y = 6 can be rewritten as y = (2/3)x - 2. The slope is 2/3.

Among these, the equation 2x - 3y = 6 has a different slope (2/3), so the system formed by this equation and any of the other equations will have exactly one solution.

Answer 2

Answer:To determine which equation results in a system of equations with exactly one solution when graphed on the same coordinate grid, we need to analyze the equations and their slopes.

When two linear equations have different slopes, they will intersect at exactly one point, resulting in a unique solution. If the slopes are the same, the lines will either be parallel and have no solution or coincident and have infinitely many solutions.

Let's examine the given equations:

1. 2x - y = 1

2. 4x - 2y = -4

3. 4x - 2y = -2

4. 2x - 3y = 6

To determine the slopes of these equations, we can rewrite them in slope-intercept form (y = mx + b), where "m" represents the slope:

1. y = 2x - 1 (slope = 2)

2. y = 2x + 2 (slope = 2)

3. y = 2x + 1 (slope = 2)

4. y = (2/3)x - 2 (slope = 2/3)

From the analysis, we can see that the equation with a unique slope is equation 4: 2x - 3y = 6. Its slope is 2/3, which is different from the slopes of the other equations. Therefore, when graphed on the same coordinate grid, the system of equations represented by equation 4 will have exactly one solution.

Explanation: