Finding the Distance between places

Question

The total distance from Austin to Dallas and back to Houston is about 600 miles. The distance from Austin to Houston is 34 miles less than the distance from Austin to Dallas. The distance from Houston to Dallas is 77 miles greater than the distance from Houston to Austin.

Answer 1

The total distance from Austin to Dallas and back to Houston is about 600 miles. The distance from Austin to Houston is 34 miles less than the distance from Austin to Dallas. The distance from Houston to Dallas is 77 miles greater than the distance from Houston to Austin.

So the total distance from Austin to Dallas, back to Houston is 600 miles.

D_AD + D_DH = 600

The distance from Austin to Houston is 34 miles less than the distance from Austin to Dallas.

D_AH = D_AD - 34

The distance from Houston to Dallas is 77 miles greater than the distance from Houston to Austin.

D_HD = D_HA + 77

So now we have a system of equations,

X, is the distance between Austin and Dallas.

Y, is the distance between Dallas and Houston.

Z, is the distance between Austin and Houston.

Rewriting the equations above we have

X + Y = 600

Z = X - 34

Y = Z + 77

If we start with the first equation

X + Y = 600 we see the third equation has a value for Y, Y=Z+77

X + (Z + 77) = 600 we see the second equation has a value for Z, Z=X-34

X + ((X-34) + 77) = 600 we can now combine like terms

2X + 43 = 600

2X = 557

X = 278.5 miles, the distance between Austin and Dallas

Starting with our Second Equation

Z = X - 34 we can use our calculated value for X

Z = 278.5 - 34

Z = 244.5 miles, distance between Austin and Houston

And finally with our third equation

Y = Z + 77 we can use our calculated value for Z

Y = 244.5 + 77

Y = 321.5 miles, the distance between Dallas and Houston