Question

For the lines $2x+5y=7$ and $2x+5y=9,$ which of the following statements is true?

• A. Lines are parallel
• B. Lines are coincident
• C. Lines are intersecting
• D. Lines are perpendicular

Answer 1

The correct option is A

Lines are parallel

Explanation for the correct option:

Step1.Given Equations:

The given equations of the lines are $2x+5y=7$and $2x+5y=9$

$\Rightarrow 5y=-2x+7$ and $\Rightarrow 5y=-2x+9$

$\Rightarrow y=\frac{-2}{5}x+\frac{7}{5}......\left(i\right)$ and

$\Rightarrow y=\frac{-2}{5}x+\frac{9}{5}.......\left(ii\right)$

We know that

The general equation of a line is $\mathit{y}\mathbf{=}\mathit{m}\mathit{x}\mathbf{+}\mathit{c}$

where, $m$ is the slope.

Step2. Find the slopes:

Let $m_1$and $m_2$ are the slope of the lines (i) and (ii).

So, that

The slope of line (i) is $m_1=-\frac{2}{5}$

The slope of line (ii) is $m_2=-\frac{2}{5}$

Thus ${\mathit{m}}_{\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{}{\mathit{m}}_{\mathbf{2}}$

Since the slopes are the same.

So the lines are parallel.

Step-3 : Explanation for incorrect options:

Option(B) Lines are coincident

We have, $2x+5y-7=0$ and $2x+5y-9=0$

Here, $a_1=2,b_1=5,c_1=-7$ and $a_2=2,b_2=5,c_2=-9$

Now, $\frac{a_1}{a_2}=\frac{2}{2},\frac{b_1}{b_2}=\frac{5}{5},\frac{c_1}{c_2}=\frac{-7}{-9}$

$\Rightarrow$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

So, the lines are not coincident.

Option(C) Lines are intersecting

Solve equation(i) and (ii)

$\frac{-2}{5}x+\frac{7}{5}=\frac{-2}{5}x+\frac{9}{5}$

$\frac{7}{5}=\frac{9}{5}$

hence, there is no real value of $x$

So, the lines are not intersecting.

Option(D) Lines are Perpendicular

we have, $m_1=-\frac{2}{5}$ and $m_2=-\frac{2}{5}$

Now, $m_1{m}_2=-\frac{2}{5}\times -\frac{2}{5}$

$\ne -1$

So, the lines are not perpendicular.

Hence, Option (A) is the correct answer.