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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?

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2 Answers

There is no common solution to these two equations. There are a number of ways to explain why.

One way is to notice that the left hand sides of both equations are equal (both are just x).
 Therefore the right hand sides must be equal. But then you have 4y + 1 = 4y - 1. Cancelling the 4y on both sides gives 1 = -1 which is clearly false.


Another way to see it is that the lines have the same slope but different intercepts and are therefore parallel. Two parallel lines never intersect so there is no common solution.

They do not. The last operation in each function (add one; and, subtract one) are different, creating different results for solutions.

i.e. if you select begin with y = 0, then the solutions for each are as follows:

x = 4y + 1                           x = 4y - 1

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

x = 0 + 1                            x = 0 - 1

x = 1                                 x = -1

 

You can repeat this for any values of y and you will find that the values of x in the first equation will always be 2 higher than the values on the second equation.

 

PROVING INEQUALITY THROUGH SUBSTITUTION

If you wish to prove equality using an operator. Set the equations equal to each other as such:

       x = x

4y + 1 = 4y - 1

-4y         -4y                Subtract 4y from each side

1        =   -1

 

Because 1 does not equal -1 you can conclude that the results from each equation will never match.