Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Liz is training for a triathlon and needs to swim a certain distance fo... Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Liz is training for a triathlon and needs to swim a certain distance for today's workout. If she swims at the rec center pool, she will complete a 148-yard warmup and then swim laps in a lane that is 50 yards long. If Liz swims at the indoor pool at her gym, she will complete a 296-yard warmup, plus a main set that consists of 46 yards per lap. If Liz swims the correct number of laps, she can complete the same distance in either pool. How many laps will that take? How long will Liz's workout be in total? If Liz swims ___ laps at either pool, she can complete a total workout of ___ yards. Read more

Steps to Solve

1. Define Variables

Let ( x ) be the number of laps Liz swims at the rec center pool.
Let ( y ) be the number of laps Liz swims at the indoor pool.

2. Create the Equation for the Rec Center Pool

Liz's total distance for the rec center pool includes a 148-yard warmup plus the distance from swimming ( x ) laps of 50 yards each:
$$ \text{Distance}_{\text{rec}} = 148 + 50x $$

3. Create the Equation for the Indoor Pool

Liz's total distance for the indoor pool includes a 296-yard warmup plus the distance from swimming ( y ) laps of 46 yards each:
$$ \text{Distance}_{\text{indoor}} = 296 + 46y $$

4. Set the Equations Equal

Since Liz swims the same distance in both pools, we can set the two total distances equal to each other:
$$ 148 + 50x = 296 + 46y $$

5. Express ( y ) in terms of ( x ) or vice versa

Rearranging the equation to solve for ( y ):
$$ 50x - 46y = 148 - 296 $$
This simplifies to:
$$ 50x - 46y = -148 $$

6. Choose a second relation for another variable

If Liz swims the same number of laps in both pools, we can assume ( x = y ):
$$ y = x $$

7. Substitute ( y ) in the distance equation

Substituting ( y ) into the rearranged equation gives:
$$ 50x - 46x = -148 $$ This simplifies to:
$$ 4x = -148 $$

8. Solve for ( x )

Isolating ( x ):
$$ x = \frac{-148}{4} = -37 $$
Since this does not make sense in the real context, we go back and realize our assumption.

Instead, let's establish another equation.

9. Establish positive distances

We revert to the earlier setup and notice:

From ( 50x = 148 + 46y - 296 ), rechecking the positive relationship will aid.

10. Count total distance after correction

By letting ( y = x ):
$$ 50x + 148 = 296 + 46x
$$
Thus simplifying yields solutions once substituted correctly.