ANSWER
She averaged at 32 mph for 5 hours and 39 mph for 2 hours.
EXPLANATION
Let
[tex]x[/tex]
be the number of hours she travelled before noon and
[tex]y[/tex]
be the number of hours she travelled after noon.
Then since she travelled for a total of 7 hours, we can write the equation,
[tex]x + y = 7.....eqn(1)[/tex]
We were also given that, before noon,she averaged at 32 mph.
We know that,
[tex]speed = \frac{distance}{time \: taken} [/tex]
Let
[tex]d_1[/tex]
be the distance before noon.
[tex]\Rightarrow \: 32 = \frac{d_1}{x} [/tex]
This implies that,
[tex]d_1 = 32x[/tex]
Also let the distance she covered after noon be,
[tex]d_2[/tex]
Then,
[tex] 39 = \frac{d_2}{y} [/tex]
This implies that,
[tex]d_2 = 39y[/tex]
Since she covered a total distance of 238 miles, we can write the equation,
[tex]d_1 + d_2 = 238[/tex]
This means that,
[tex]32x + 39y = 238....eqn(2)[/tex]
We multiply equation by 32 to get,
[tex]32x + 32y = 224...eqn(3)[/tex]
Equation (2) minus equation (3), will give us,
[tex]7y = 14[/tex]
This implies that,
[tex]y = 2[/tex]
We substitute the value of y into equation (1) to get,
[tex]x + 2 = 7[/tex]
[tex]x = 7 - 2[/tex]
[tex]x = 5[/tex]
We can now conclude that she averaged at 32 mph for 5 hours and 39 mph for 2 hours.
