On the basis of data from the years 2000 through 2015 , seasonally adjusted median weekly earnings $$ y $$ in year $$ x $$ for men and women in a given area can be approximated by the following equations, where $$ x=0 $$ corresponds to the year 2000 and $$ y $$ is in current dollars. If these equations remain valid in the future, will earnings for women catch up with men within the next decade? Women: $$ \quad-793 x+50 y=24,800 $$ Men: $$ \quad-88 x+5 y=3,230 $$ $$ \begin{array}{l} ext { A. Yes. The equations intersect at a positive } x ext {-value, which is after the year } 2015 ext {. This indicates that the earnings for men and women will be the same in the next decade. } \ ext { Choose the correct answer below. } \ ext { B. No. The equations intersect at a negative } x ext {-value, which is before the year } 2000 ext {. However, since the equations are linear, they cannot share another intersection point. } \ ext { This indicates that the earnings for men and women will not be the same in the next decade. } \ ext { C. Yes. The equations indicate that the earnings for men and women are ahvays the same } \ ext { D. No. The equations indicate that the earnings for men and women never were and never will be the same. }\end{array} $$

Question

On the basis of data from the years 2000 through 2015 , seasonally adjusted median weekly earnings $$ y $$ in year $$ x $$ for men and women in a given area can be approximated by the following equations, where $$ x=0 $$ corresponds to the year 2000 and $$ y $$ is in current dollars. If these equations remain valid in the future, will earnings for women catch up with men within the next decade? Women: $$ \quad-793 x+50 y=24,800 $$ Men: $$ \quad-88 x+5 y=3,230 $$ $$ \begin{array}{l}	ext { A. Yes. The equations intersect at a positive } x 	ext {-value, which is after the year } 2015 	ext {. This indicates that the earnings for men and women will be the same in the next decade. } \ 	ext { Choose the correct answer below. } \ 	ext { B. No. The equations intersect at a negative } x 	ext {-value, which is before the year } 2000 	ext {. However, since the equations are linear, they cannot share another intersection point. } \ 	ext { This indicates that the earnings for men and women will not be the same in the next decade. } \ 	ext { C. Yes. The equations indicate that the earnings for men and women are ahvays the same } \ 	ext { D. No. The equations indicate that the earnings for men and women never were and never will be the same. }\end{array} $$

UpStudy AI · Accepted Answer

We start with the given equations:

For women:
$$
-793x + 50y = 24,\!800
$$
For men:
$$
-88x + 5y = 3,\!230
$$

Since we want to know if earnings for women catch up with men (i.e. when the equations have the same $$ y $$) within the next decade (i.e. for $$ x > 0 $$), we find the intersection of the two lines.

**Step 1. Express each equation in terms of $$ y $$:**

For women:
$$
50y = 24,\!800 + 793x \quad \Longrightarrow \quad y = \frac{24,\!800 + 793x}{50}
$$

For men:
$$
5y = 3,\!230 + 88x \quad \Longrightarrow \quad y = \frac{3,\!230 + 88x}{5}
$$

**Step 2. Set the two expressions for $$ y $$ equal:**

$$
\frac{24,\!800 + 793x}{50} = \frac{3,\!230 + 88x}{5}
$$

**Step 3. Clear the denominators by multiplying both sides by 50:**

$$
24,\!800 + 793x = 10 \times (3,\!230 + 88x)
$$

Simplify the right-hand side:

$$
24,\!800 + 793x = 32,\!300 + 880x
$$

**Step 4. Solve for $$ x $$:**

Subtract $$ 793x $$ from both sides:
$$
24,\!800 = 32,\!300 + 87x
$$

Subtract 32,\!300 from both sides:
$$
24,\!800 - 32,\!300 = 87x
$$
$$
-7,\!500 = 87x
$$

Divide by 87:
$$
x = -\frac{7,\!500}{87}
$$

Since $$ -\frac{7,\!500}{87} $$ is negative, the intersection occurs at a negative $$ x $$-value, which corresponds to a year before 2000 (recall $$ x = 0 $$ corresponds to the year 2000).

**Conclusion:**

Since the two lines intersect at a negative $$ x $$-value and linear equations only intersect at one point, there is no future point (i.e. $$ x > 0 $$) at which women’s earnings will catch up with men’s earnings.

Thus, the correct answer is:

B. No. The equations intersect at a negative $$ x $$-value, which is before the year 2000. However, since the equations are linear, they cannot share another intersection point. This indicates that the earnings for men and women will not be the same in the next decade.
The equations intersect at a negative value of $$ x $$, meaning women's earnings won't catch up to men's in the next decade. So, the answer is **B. No**.

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