Question

3 multiple choice 2 points
paul swims and bikes for two weeks. he swims x miles per hour and bikes y miles per hour. the system shows the total number of miles he exercised for the two weeks.
\

$$\begin{cases} 2x + 3.5y = 45 \\\\ 4x + 7y = 100 \\end{cases}$$

how many solutions does the system have?
\bigcirc two solutions
\bigcirc one solution
\bigcirc no solution
\bigcirc infinitely many solutions

Explanation:

Step1: Analyze the linear system

We have the system:

$$\begin{cases} 2x + 3.5y = 45 \\ 4x + 7y = 100 \end{cases}$$

Step2: Scale the first equation

Multiply the first equation by 2 to compare coefficients:
$2\times(2x + 3.5y) = 2\times45$
$\implies 4x + 7y = 90$

Step3: Compare with the second equation

The second equation is $4x + 7y = 100$. We now have two equations:
$4x + 7y = 90$ and $4x + 7y = 100$
These represent parallel lines (same coefficients for $x$ and $y$, different constants), which never intersect.

Answer:

no solution