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 first need to understand what the equations mean. In both equations, x represents the value of a variable and y represents another variable. The equations express a relationship between x and y.
In the first equation, x = 4y + 1, we can see that for any value of y, if we multiply it by 4 and then add 1, we will get the corresponding value of x. For example, if y = 2, then x = 4(2) + 1 = 9.
Similarly, in the second equation, x = 4y – 1, we can see that for any value of y, if we multiply it by 4 and then subtract 1, we will get the corresponding value of x. For example, if y = 2, then x = 4(2) – 1 = 7.
Now, let's compare the two equations. If we ignore the constants (1 and -1 in this case), we can see that both equations are of the form x = 4y. This means that for any value of y, the value of x will be four times that value.
To determine if they have the same solution, we need to consider the values of x and y that satisfy both equations. We can do this by finding a common value for x and y that satisfies both equations, if such a value exists.
If we subtract the two equations, we get:
(x = 4y + 1) - (x = 4y - 1)
0 = 2
Here, we see that this equation simplifies to 0 = 2, which is not true. This means that there is no common solution to both equations, and therefore, x = 4y + 1 and x = 4y – 1 do not have the same solution.
To explain this to someone who has not learned algebra, you can use a visual representation. By graphing both equations on a coordinate plane (with x on the x-axis and y on the y-axis), you will see that the lines representing the equations are parallel. This means they never intersect and therefore do not share any common solutions.