Question

the table below shows the percentage of male and female populations in a certain country employed in the civilian work force in selected years from 1970 to 2005. algebraically, the linear model for the female data is y1 = 0.493x + 20.9 and the linear model for the male data is y2 = - 0.683x + 84.1, where the 1970 and 2000 ordered pairs are used to compute the slopes. if the percentages continue to follow the linear models, at what point will the lines intersect? what is the significance of the intersection point in terms of employment participation?

year	1970	1975	1980	1985	1990	1995	2000	2005
female (%)	20.9	31.2	33.5	38.6	40.8	42.3	43.7	42.1
male (%)	84.1	80.6	78.7	75.2	68.3	65.7	63.6	65.5

the lines will intersect at the point
(type an ordered pair. round to the nearest tenth as needed.)

Explanation:

Step1: Set the two linear - equations equal

We have the female data model $y_1 = 0.493x+20.9$ and the male data model $y_2=- 0.683x + 84.1$. Set $y_1=y_2$, so $0.493x+20.9=-0.683x + 84.1$.

Step2: Combine like - terms

Add $0.683x$ to both sides of the equation: $0.493x + 0.683x+20.9=-0.683x+0.683x + 84.1$. This simplifies to $1.176x+20.9 = 84.1$.

Step3: Isolate the variable $x$

Subtract 20.9 from both sides: $1.176x+20.9 - 20.9=84.1 - 20.9$. We get $1.176x=63.2$. Then divide both sides by 1.176: $x=\frac{63.2}{1.176}\approx53.8$.

Step4: Find the value of $y$

Substitute $x\approx53.8$ into the equation $y = 0.493x+20.9$. So $y=0.493\times53.8+20.9$. First, $0.493\times53.8 = 0.493\times(54 - 0.2)=0.493\times54-0.493\times0.2=(0.5 - 0.007)\times54-(0.5 - 0.007)\times0.2=(27-0.378)-(0.1 - 0.0014)=26.622 - 0.0986 = 26.5234$. Then $y=26.5234+20.9 = 47.4234\approx47.4$.

Answer:

$(53.8,47.4)$