Question

which system has infinitely many solutions?
$y = 2x + 1,$
$y = 2x - 3$
$y = 3x - 1,$
$y = -x + 5$
$y = x + 2,$
$y = -x + 2$
$y = x, y = x$

Explanation:

Step1: Recall the condition for infinitely many solutions

A system of linear equations \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \) has infinitely many solutions if they are the same line, i.e., \( m_1 = m_2 \) and \( b_1 = b_2 \).

Step2: Analyze each system

• First system: \( y = 2x + 1 \) and \( y = 2x - 3 \). Here, \( m_1 = 2, b_1 = 1 \); \( m_2 = 2, b_2 = -3 \). \( b_1

eq b_2 \), so no infinite solutions.

• Second system: \( y = 3x - 1 \) and \( y = -x + 5 \). \( m_1 = 3, m_2 = -1 \), \( m_1

eq m_2 \), so no infinite solutions.

• Third system: \( y = x + 2 \) and \( y = -x + 2 \). \( m_1 = 1, m_2 = -1 \), \( m_1

eq m_2 \), so no infinite solutions.

• Fourth system: \( y = x \) and \( y = x \). Here, \( m_1 = 1, b_1 = 0 \); \( m_2 = 1, b_2 = 0 \). So \( m_1 = m_2 \) and \( b_1 = b_2 \), meaning they are the same line.

Answer:

The system \( y = x, y = x \) has infinitely many solutions.