Final answer:
A linear equation that has no solution when paired with y = 2x + 1 is y = 2x + 3. This is because both equations have the same slope but different y-intercepts, indicating that they are parallel and do not intersect. The algebraic proof by setting the equations equal to each other results in a contradiction, demonstrating that there is no solution.
Step-by-step explanation:
To answer the student's question, we need to find a linear equation that when paired with y = 2x + 1 has no solution. A pair of linear equations has no solution if they are parallel, which means they have the same slope but different y-intercepts. The given equation has a slope of 2. Therefore, another equation that is parallel to y = 2x + 1 would have the same slope of 2 but a different y-intercept.
Let's consider the equation 4x + 2y = 10. We can rewrite it in slope-intercept form (y = mx + b) as follows:
- Divide the entire equation by 2 to isolate the y-term: 2x + y = 5
- Subtract 2x from both sides to solve for y: y = -2x + 5
Now we have the equation in the form y = -2x + 5, which clearly shows that the slope is -2, not 2. Thus, this is not the equation we are looking for. We need one with a slope of 2.
Instead, let's consider the equation y = 2x + 3. This equation has a slope of 2, which is the same as the slope in the initial equation y = 2x + 1, but the y-intercept is different (3 vs 1). Since they have the same slope but different y-intercepts, they are parallel and will never intersect, meaning there is no solution to this system of equations.
Algebraic proof that there is no solution:
- y = 2x + 1 (given)
- y = 2x + 3 (equation with no solution)
- If you set these equal to each other (as you would do to find a point of intersection), you would get: 2x + 1 = 2x + 3
- Subtract 2x from both sides: 1 = 3, which is a contradiction.
Since we arrive at a false statement, this confirms that there is no solution for the system of equations y = 2x + 1 and y = 2x + 3.