Question

consider the system of equations.\

$$\begin{cases} y = 2x + 2 \\\\ y = 6x + 2 \\end{cases}$$

which statements are true about the system of equations?select each correct answer.a. the graph of the system consists of lines that have no points of intersection.b. the graph of the system consists of lines that have exactly one point of intersection.c. the graph of the system consists of lines that have more than one point of intersection.d. the system has no solution.e. the system has exactly one solution.f. the system has more than one solution.\boxed{a} \boxed{b} \boxed{c} \boxed{d} \boxed{e} \boxed{f}

Explanation:

Step1: Analyze the slopes and y-intercepts

The two equations are in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
For the equation \(y=2x + 2\), the slope \(m_1=2\) and the y - intercept \(b_1 = 2\).
For the equation \(y = 6x+2\), the slope \(m_2=6\) and the y - intercept \(b_2=2\).

Since the slopes \(m_1
eq m_2\) ( \(2
eq6\)) and the y - intercepts \(b_1 = b_2=2\), the two lines are not parallel (because parallel lines have equal slopes) and they are not the same line (because if they were the same line, both slope and y - intercept would be equal). So, the two lines will intersect at exactly one point.

Step2: Analyze the number of solutions

A system of linear equations \(y = m_1x + b_1\) and \(y=m_2x + b_2\) has:

• No solution if the lines are parallel ( \(m_1=m_2\) and \(b_1

eq b_2\))

• Exactly one solution if the lines intersect at one point ( \(m_1

eq m_2\))

• Infinitely many solutions if the lines are the same ( \(m_1=m_2\) and \(b_1 = b_2\))

Since \(m_1 = 2\), \(m_2=6\) ( \(m_1
eq m_2\)), the system has exactly one solution. And the graph of the system (the two lines) has exactly one point of intersection.

Answer:

B. The graph of the system consists of lines that have exactly one point of intersection.
E. The system has exactly one solution.