Based on data from 1997 through 2010, in Canada, women's earnings as a percentage of men's can be approximated by the linear equation
y = 0.42x + 68.8
where x = the number of years since 1997
(x = 0
for 1997) and y = the percentage. Based on this equation, find the approximate percentage of women's earnings for the following years.
When will the percentage reach 90%

To solve this problem, we'll use the given linear equation:

\[ y = 0.42x + 68.8 \]

where:
- \( y \) represents the percentage of women's earnings as a percentage of men's earnings.
- \( x \) represents the number of years since 1997.

### Part 1: Calculate the Percentage for Specific Years

Let's calculate \( y \) for each year.

1. **For the year 2000**:
   - \( x = 2000 - 1997 = 3 \)
   - \( y = 0.42(3) + 68.8 = 1.26 + 68.8 = 70.06 \% \)

2. **For the year 2005**:
   - \( x = 2005 - 1997 = 8 \)
   - \( y = 0.42(8) + 68.8 = 3.36 + 68.8 = 72.16 \% \)

3. **For the year 2010**:
   - \( x = 2010 - 1997 = 13 \)
   - \( y = 0.42(13) + 68.8 = 5.46 + 68.8 = 74.26 \% \)

### Part 2: Determine When the Percentage Reaches 90%

To find when \( y \) reaches 90%, we set \( y = 90 \) and solve for \( x \):

\[ 90 = 0.42x + 68.8 \]

Subtract 68.8 from both sides:

\[ 90 - 68.8 = 0.42x \]

\[ 21.2 = 0.42x \]

Now, solve for \( x \):

\[ x = \frac{21.2}{0.42} \approx 50.48 \]

Since \( x \) represents the number of years since 1997:

\[ \text{Year} = 1997 + 50.48 \approx 2047.48 \]

Thus, the percentage will reach approximately 90% around the year **2047**.

### Summary
- The percentage of women's earnings as a percentage of men's earnings was approximately:
  - 70.06% in 2000
  - 72.16% in 2005
  - 74.26% in 2010
- The percentage is expected to reach 90% around the year 2047.

Would you like further details or have any questions?

Here are 5 questions that might help expand on this:

1. How do we interpret the slope (0.42) in this context?
2. What would the percentage have been in 1997 according to the equation?
3. Can this linear model be used indefinitely? Why or why not?
4. How would changes in the slope affect the year the percentage reaches 90%?
5. What other factors might influence the accuracy of this prediction?

**Tip:** When solving linear equations, always check if the values for \( x \) and \( y \) make sense within the context of the problem, particularly when extending the model into the future.

Learn how to calculate women's earnings as a percentage of men's using the linear equation y = 0.42x + 68.8 based on data from 1997 through 2010 in Canada. Understand when the percentage will reach 90% and explore the implications of the linear model. Suitable for high school students studying linear equations.

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