Solve the equation: -1/7 x = 5. Then, solve -1/7 x : (-1/7) = 5 : (-1/7).

Hello! Let's break down this math problem step-by-step.

Part 1: Solve for x in the equation \[ -\frac{1}{7}x = 5 \]

To isolate x, we need to get rid of the fraction -1/7. We can do this by multiplying both sides of the equation by the reciprocal of -1/7, which is -7.

1. Multiply both sides by -7:
   \[ \left(-\frac{1}{7}x\right) \times (-7) = 5 \times (-7) \]
2. Simplify:
   \[ x = -35 \]

Part 2: Solve the expression \[ -\frac{1}{7}x : \left(-\frac{1}{7}\right) = 5 : \left(-\frac{1}{7}\right) \]

We already know that x = -35. Let's substitute this value into the left side of the equation.

1. Substitute x = -35:
   \[ \left(-\frac{1}{7}\right) \times (-35) : \left(-\frac{1}{7}\right) \]
2. First, calculate \( \left(-\frac{1}{7}\right) \times (-35) \). This is the same as dividing -35 by 7, which gives 5.
   \[ 5 : \left(-\frac{1}{7}\right) \]
3. Now, we need to divide 5 by -1/7. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of -1/7 is -7.
   \[ 5 \times (-7) \]
4. Calculate the result:
   \[ -35 \]

Now let's look at the right side of the original equation: \( 5 : \left(-\frac{1}{7}\right) \). This is the same calculation we just did!

1. Divide 5 by -1/7:
   \[ 5 \times (-7) \]
2. Calculate the result:
   \[ -35 \]

So, both sides of the second equation equal -35.

Answer:

• The solution to \( -\frac{1}{7}x = 5 \) is \( x = -35 \).
• The solution to \( -\frac{1}{7}x : \left(-\frac{1}{7}\right) = 5 : \left(-\frac{1}{7}\right) \) is \( -35 \).