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# Do the equations x = 4y + 1 and x = 4y – 1 have the same solution? How might you explain your answer to someone who has not learned algebra?

Do the equations x = 4y + 1 and x = 4y – 1 have the same solution? How might you explain your answer to someone who has not learned algebra?

No, they do not have the same solution. The difference is that in one you are ADDING 1 and in the other you are SUBTRACTING 1.

In order to solve equations with two variables (x and y) you have to pick a value for one of the variables in order to cancel it out and only have one variable. To make it easy, we will make x=0.

Now we have 0 = 4y + 1

When solving algebra you always try to get the variable alone. We will be trying to get "y" alone by getting rid of the 1 and 4. We do this by doing the opposite function of which they are in right now. In this problem we have + 1 so we will get rid of it by subtracting it, and what you do to one side of the "=" you will do to the other side. We will get rid of the 4 by dividing it. We do this because right now we have 4y (4 TIMES y), the opposite of multiplication is division. So:

0 = 4y + 1

-1           -1

-1 = 4y

/4    /4

-1/4 = y

now we know that y = -1/4 and that x = 0. If you want to check your answers, substitute both variables for the numbers obtained:

(0) = 4(-1/4) + 1 0 = -1 + 1 0 = 0

The equation x = 4y - 1 does not have the same solution because the one is being subtracted rather than added. Let's look at the steps. We will make x=0 again.

0 = 4y - 1

+1 + 1

1 = 4y

/4 /4

1/4 = y

Our answer for "y" is 1/4 in this problem compared to the -1/4 in the last one. Do you see how the difference between the subtraction or addition sign matters?

Solve both for y:

1)
x = 4y + 1
x - 1 = 4y
(1/4)x - 1/4 = y
or
y = (1/4)x - 1/4

slope = 1/4
y intercept = (0, -1/4)

2)
x = 4y – 1
x + 1 = 4y
(1/4)x + 1/4 = y
or

y = (1/4)x + 1/4

slope = 1/4
y intercept = (0, 1/4)

When two linear equations have the same slope, they are parallel (the do not intersect) and thus have no common solution.

Hope this helps.
Jason