What relation do you see between the following lines: 1) y = 3x + 1 and y = 3x + 5, 2) y = −2/3 x + 1 and y…

Question

What relation do you see between the following lines: 1) y = 3x + 1 and y = 3x + 5, 2) y = −2/3 x + 1 and y = 2/3 x + 4

ScanSolve AI Tutor · Accepted Answer

To determine the relation between lines, we compare their slopes. The general form of a linear equation is $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.

If two lines have the same slope ( $m_1 = m_2$ ), they are parallel.
If the product of their slopes is $-1$ ( $m_1 \cdot m_2 = -1$ ), they are perpendicular.

1) For the lines $y = 3x + 1$ and $y = 3x + 5$ :

Identify the slopes of the two lines.
The first line is $y = 3x + 1$ , so its slope $m_1 = 3$ .
The second line is $y = 3x + 5$ , so its slope $m_2 = 3$ .
Compare the slopes.
Since $m_1 = 3$ and $m_2 = 3$ , the slopes are equal ( $m_1 = m_2$ ).
State the relationship.
The lines are parallel.
$\boxed{\text{Parallel}}$
2) For the lines $y = -\frac{3}{2}x + 1$ and $y = \frac{2}{3}x + 4$ :
Identify the slopes of the two lines.
The first line is $y = -\frac{3}{2}x + 1$ , so its slope $m_1 = -\frac{3}{2}$ .
The second line is $y = \frac{2}{3}x + 4$ , so its slope $m_2 = \frac{2}{3}$ .
Calculate the product of the slopes.
$m_1 \cdot m_2 = \left(-\frac{3}{2}\right) \cdot \left(\frac{2}{3}\right)$ $m_1 \cdot m_2 = -\frac{3 \times 2}{2 \times 3}$ $m_1 \cdot m_2 = -\frac{6}{6}$ $m_1 \cdot m_2 = -1$
State the relationship.
Since the product of their slopes is $-1$ , the lines are perpendicular.
$\boxed{\text{Perpendicular}}$

What relation do you see between the following lines: 1) y = 3x + 1 and y = 3x + 5, 2) y = -2/3 x + 1 and y = 2/3 x + 4

Step-by-step solution

Identify the slopes of the two lines.

Compare the slopes.

State the relationship.

Identify the slopes of the two lines.

Calculate the product of the slopes.

State the relationship.

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