Question 179569
I will write 2 equations:
1 for running
1 for walking
(1) {{{d[w] = r[w]*t[w]}}}
(2) {{{d[r] = r[r]*t[r]}}}
given:
{{{r[w] = 70}}} m/min
{{{r[r] = 210}}} m/min
{{{d[w] + d[r] = 1800}}} m
{{{t[w] + t[r] <= 20}}}min
--------------------------
I can rewrite (1) and (2):
(1) {{{d[w] = 70t[w]}}} m
(2) {{{d[r] = 210t[r]}}} m
Now, from
{{{d[w] + d[r] = 1800}}}  I get
{{{d[r] = 1800 - d[w]}}}
I'll substitute this in (2)
(2) {{{1800 - d[w] = 210t[r]}}} m
{2) {{{d[w] = 1800 - 210t[r]}}} m
I'll subtract (1) from (2):
{2) {{{d[w] = 1800 - 210t[r]}}}
(1) {{{-d[w] = -70t[w]}}}
{{{0 = 1800 - 210t[r] - 70t[w]}}}
{{{70t[w] + 210t[r] = 1800}}}
Divide both sides by {{{10}}}
(3) {{{7t[w] + 21t[r] = 180}}}
from 
{{{t[w] + t[r] <= 20}}}min
If {{{t[w] + t[r] = 20}}}min
Multiply each side by {{{7}}}
(4) {{{7t[w] + 7t[r] = 140}}}min
Subtract (4) from (3)
(3) {{{7t[w] + 21t[r] = 180}}}
(4) {{{-7t[w] - 7t[r] = -140}}}
{{{14t[r] = 40}}}
{{{t[r] = 20/7}}} min
and,since
{{{t[w] + t[r] = 20}}}min
{{{t[w] = 20 - t[r]}}}
{{{t[w] = 20 - 20/7}}}
{{{t[w] = (140 - 20)/7}}}
{{{t[w] = 120/7}}} min
The problem wants to know how far he ran
(2) {{{d[r] = 210t[r]}}} m
{{{d[r] = 210*(20/7)}}} m
{{{d[r] = 600}}} m
He ran 600 m
check answer:
I can rewrite (1) and (2):
(1) {{{d[w] = 70t[w]}}} m
{{{d[w] = 70*(120/7)}}} m
{{{d[w] = 1200}}} m
And, since
{{{d[w] + d[r] = 1800}}} m
{{{600 + 1200 = 1800}}}
{{{1800 = 1800}}}
OK
I also looked at 
{{{t[w] + t[r] <= 20}}}min
If he spent a little more time running than
walking, say
{{{t[r] = 21/7}}}
and
{{{t[w] = 119/7}}}
{{{d[w] + d[r] = 1820}}} m
If he spent a little more time walking than running
{{{t[r] = 19/7}}}
{{{t[w] = 121/7}}}
{{{d[w] + d[r] = 1780}}} m
So, he has to spend at least as much time running
as he did