Maria drove a total of 280 miles. She drove for x hours at 35 mph and y hours at 70...

1. TRANSLATE the problem information

• Given information:
  • Maria drove 280 miles total
  • She drove x hours at 35 mph
  • She drove y hours at 70 mph
  • Equation: \(35x + 70y = 280\)
• What this tells us: The equation must represent how the total distance breaks down by speed

2. INFER what the equation structure means

• The equation shows total distance (280) equals the sum of two terms
• Each term must represent distance traveled at each speed
• Distance = Speed × Time, so each term follows this pattern

3. TRANSLATE each term in the equation

• \(35x = 35\) mph \(\times\) \(x\) hours = distance traveled at 35 mph
• \(70y = 70\) mph \(\times\) \(y\) hours = distance traveled at 70 mph
• \(35x + 70y =\) total distance \(= 280\) miles

4. Identify what 70y represents

• \(70y =\) speed \(\times\) time = distance
• Specifically: \(70y =\) distance driven at 70 mph

Answer: C - The total miles driven at 70 mph

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students see \(70y\) and focus only on individual parts rather than understanding the complete expression.

They might think "70 is the speed and y is the time, so 70y must be asking about either the speed or the time." This leads them to select Choice A (The number of hours driven at 70 mph) because they see y as representing hours, or they get confused between the components.

Second Most Common Error:

Missing conceptual knowledge about distance formula: Students don't automatically recognize that when speed and time are multiplied together, the result is distance.

Without this connection, they may interpret \(70y\) as something other than distance, leading to confusion about what the expression actually represents. This causes them to get stuck and guess among the choices.

The Bottom Line:

Success requires recognizing that algebraic expressions in word problems represent meaningful quantities - here, that multiplying speed by time gives you the distance traveled at that speed. The key insight is seeing \(70y\) as one complete unit (distance) rather than two separate parts (speed and time).