Question

Please help me! 90 points!

The equation shown below has exactly one solution.

−5x + 3 = 2(x + 1)

Step 1: Change the equation so it has no solutions.

Step 2: Change your equation from Step 1 so it has infinitely many solutions.

Rules
1. All equations must be linear and have only variable, x

2. You can make one—and only one—of the following changes to each equation:
o Change one term.
o Delete one term.
o Add one term.

What are your equations?

Explain how you know each equation has one, none, or infinitely many solutions.

Answer 1

1/7

Explanation:

Answer 2

To make it so that the equations have no solutions, we have to make the x's cancel out.

Lets do this by changing -5 into 2.

Equation 1;

`2x+3=2(x+1)`

This works, because when we expand the brackets, we get 2x + 1 on the right side. Then when we subtract 2x from both sides to try and solve the equation, we end up cancelling out all of the x's, leaving us with 3=2. And since 3 does not equal 2, hence there are no solutions

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To make an equation have an infinite amount of solutions, we have to make both sides (the left and right) be exactly the same.

We can do this by changing the 3 in our equation above into 2:

Equation 2:

`2x+2=2(x+1)`

No matter what we say what x is, both sides of the equation will be the same, and cancel down to 0.

In that equation, x could be 0, or 1 or 2 or 324 or 2041 or any other number (hence any infinite number of solutions)

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Explanation for the equation with only one solution:

It only has one solution, because there is only one value for x that makes the equal sign correct..

-5x + 3 = 2(x + 1)

If we solve it we get that x = 1/7.

If we substitute in 1/7 into x, we get:

-5(1/7) + 3 = 2(1/7 + 1)

-5/7 + 3 = 2/7 +2

16/7 = 16/7 (now the equal sign is correct)

But if we substitute in other number, like 3, the equal sign won't be true.

1/7 is the only number that can be substituted into x to make the equation 'correct'. Hence, there is only one solution