Do the equations x = 4y + 1 and x = 4y – 1 have the same solution? Justify your answer with an explanation or a graph. How might you explain your answer to someone who has not learned algebra?

I wouldn't try to explain it to someone who has not learned algebra. The use of variables and linear equations would be unfamiliar.

However, you can pick any point, say (1,0) on the first line and show that it does not lie on the second line.

The lines are not identical, but they have the same slope. So, they are parallel, and do not intersect.

To determine if the equations x = 4y + 1 and x = 4y – 1 have the same solution, we need to compare their graphs. Let's first understand what these equations represent:

The equations x = 4y + 1 and x = 4y – 1 are both linear equations, where x and y are variables. They are written in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In these equations, the coefficient of y is 4, indicating that the slope is 4, and the constant term (+1 or -1) represents the y-intercept.

To explain to someone who hasn't learned algebra, we can start by visualizing the graphs of these equations. We can use a graphical approach to compare the two equations.

Plotting the graphs on a coordinate system, with the x-axis representing the horizontal axis and the y-axis representing the vertical axis, we can see that both equations form lines. The general equation of a line is y = mx + b, where m is the slope and b is the y-intercept.

For the equation x = 4y + 1, we can rearrange it to y = (1/4)x - 1/4. This equation has a slope of 1/4 and a y-intercept of -1/4.

For the equation x = 4y – 1, we can rearrange it to y = (1/4)x + 1/4. This equation also has a slope of 1/4, but its y-intercept is 1/4.

Now, plotting the graphs of both equations on the coordinate system, we observe that these lines are parallel. Since they have the same slope but different y-intercepts, they will never intersect. Therefore, these equations do not have the same solution.

To explain this observation to someone who hasn't learned algebra, you can use the visual comparison of the graphs. You can explain that when lines are parallel, they never intersect or touch each other. In this case, the two lines never cross, indicating that they never have the same solution.

This graphical explanation helps in understanding the concept without relying solely on algebraic comprehension.