Sure, let's work through the problem step-by-step.
1. Define variables:
- Let [tex]\( x \)[/tex] be the kilometers traveled by bike.
- Let [tex]\( y \)[/tex] be the kilometers traveled by bus.
2. Set up the equations based on the problem statements:
- They biked 75 kilometers more than they were bussed, so:
[tex]\[
x = y + 75
\][/tex]
- They traveled 325 kilometers in total, so:
[tex]\[
x + y = 325
\][/tex]
3. Substitute [tex]\( x \)[/tex] from the first equation into the second equation:
- [tex]\( x \)[/tex] is given as [tex]\( y + 75 \)[/tex], so replace [tex]\( x \)[/tex] in the second equation:
[tex]\[
(y + 75) + y = 325
\][/tex]
4. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[
y + 75 + y = 325 \\
2y + 75 = 325
\][/tex]
- Subtract 75 from both sides:
[tex]\[
2y = 250
\][/tex]
- Divide both sides by 2:
[tex]\[
y = 125
\][/tex]
5. Find [tex]\( x \)[/tex] using the value of [tex]\( y \)[/tex]:
[tex]\[
x = y + 75 \\
x = 125 + 75 \\
x = 200
\][/tex]
Thus, George and Carmen traveled 200 kilometers by bike.
