Question

use the system of linear equations below to answer the questions.\

$$\begin{cases}x + y = 5\\\\5x + 5y = b\\end{cases}$$

\\(a\\). find the value of \\(b\\) so that the system has an infinite number of solutions.\\(b = \square\\)

Explanation:

Step1: Recall infinite solutions condition

For a system of linear equations \(

$$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$

\), infinite solutions occur when \( \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \).

Step2: Analyze the given system

Given \(

$$\begin{cases} x + y = 5 \\ 5x + 5y = b \end{cases}$$

\), we can rewrite the second equation by factoring out 5: \( 5(x + y)=b \), or \( x + y=\frac{b}{5} \) (dividing both sides by 5, \( 5
eq0 \)).

Step3: Apply the infinite solutions condition

For infinite solutions, the two equations must be identical. So, \( x + y = 5 \) and \( x + y=\frac{b}{5} \) must be the same. Thus, \( \frac{b}{5}=5 \).

Step4: Solve for \( b \)

Multiply both sides by 5: \( b = 5\times5 = 25 \).

Answer:

\( b = 25 \)