Tag: general equation of a line

Questions Related to general equation of a line

For the equation given below, find the the slope and the y-intercept : $\displaystyle 3y=7$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\displaystyle 0 \ and \ \frac{7}{3}$

  2. $\displaystyle 0 \ and \ -\frac{7}{3}$

  3. $\displaystyle -\frac{7}{3} \ and \ 0$

  4. $\displaystyle \frac{7}{3} \ and \ 0$

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Correct Option: A
Explanation:

The equation of any straight line can be written as $ y =

mx + c $, where $m$ is its slope and $c$ is its y - intercept.
$ 3y = 7 $ can be written as $ y = \frac {7}{3} $

Comparing this equation with the standard form of the equation, we get:
$ m = 0 , c =  \frac {7}{3} $

Hence, slope of $ 3y = 7 $ is $ 0 $  and y -intercept is $   \frac {7}{3} $

$ax + by + c = 0$ does not represent an equation of line if ____.

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $a = c = 0, b \neq 0$

  2. $b = c = 0, a \neq 0$

  3. $a = b = 0$

  4. $c = 0, a \neq 0, b \neq 0 $

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Correct Option: C
Explanation:

$ax+by+c=0$ will represent the equation of line If both or one coefficient of $x$ and $y$ is not equal to $0$.

Therefore, if $a=b=0$ then it will not represent the equation of a line.

Find slope, x-intercept & y-intercept of the line 2x - 3y + 5 = 0

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\dfrac{-5}{2},\dfrac{5}{3},\dfrac{2}{3}$

  2. $\dfrac{-5}{2},\dfrac{5}{3},\dfrac{1}{3}$

  3. $\dfrac{-3}{2},\dfrac{5}{3},\dfrac{2}{3}$

  4. $\dfrac{-5}{2},\dfrac{4}{3},\dfrac{2}{3}$

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Correct Option: A

Find the slope and $y$-intercept of the line $2x + 5y = 1$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $=$ $-\dfrac{2}{5}$, $y$-intercept $=$ $\dfrac{1}{5}$

  2. slope $=$ $-\dfrac{1}{5}$, $y$-intercept $=$ $\dfrac{1}{5}$

  3. slope $=$ $-\dfrac{2}{3}$, $y$-intercept $=$ $\dfrac{1}{5}$

  4. slope $=$ $-\dfrac{2}{5}$, $y$-intercept $= $ $\dfrac{2}{5}$

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Correct Option: A
Explanation:
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $2x+5y=1$ in slope intercept form:
$2x+5y=1$
$5y=-2x+1$
$y=-\dfrac { 2 }{ 5 } x+\dfrac { 1 }{ 5 }$ 
Hence, the slope of the line $2x+2y=-2$ is $m=-\dfrac { 2 }{ 5 }$ and the $y$-intercept is $\dfrac { 1 }{ 5 }$.

Find the slope and $y$-intercept of the line $-5x + y = 5$.

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $= 5, y$-intercept $= -5$

  2. slope $= 5, y$-intercept $= -4$

  3. slope $= 5, y$-intercept $= 5$

  4. slope $= 5, y$-intercept $= -1$

Show answer
Correct Option: C
Explanation:

The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. 

Given straight line $-5x+y=5$ can be written as, $y=5x+5$
Now comparing above equation with slope-intercept form $y=mx+c$

We get, slope $=m = 5$ and $y$-intercept $=c=5$.

Hence, option C is correct.

Find the slope and $y$-intercept of the line $0.2x - y = 1.2$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $= 0.2$, $y$-intercept $= -1.2$

  2. slope $= 1.2$, $y$-intercept $= -1.2$

  3. slope $= 0.2$, $y$-intercept $= -2.2$

  4. slope $= 0.2$, $y$-intercept $= -1.3$

Show answer
Correct Option: A
Explanation:
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $0.2x-y=1.2$ in slope intercept form:
$0.2x-y=1.2$
$\Rightarrow -y=-0.2x+1.2$
$\Rightarrow y=0.2x-1.2$
Hence, the slope of the line $0.2x-y=1.2$ is $m=0.2$ and the $y$-intercept is $-1.2$.

Find the slope and $y$-intercept of the line $2x + 2y = -2$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope = 1, y-intercept $= -3$

  2. slope = -1, y-intercept $= -1$

  3. slope = 1, y-intercept $= 3$

  4. slope = 1, y-intercept $= 1$

Show answer
Correct Option: B
Explanation:
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $2x+2y=-2$ in slope intercept form:
$2x+2y=-2$
$\Rightarrow 2y=-2x-2$
$\Rightarrow y=-x-1$
Hence, the slope of the line $2x+2y=-2$ is $m=-1$ and the $y$-intercept is $-1$.

Find the slope and $y$-intercept of the line $x - y = 3$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $= 2$, $y$-intercept $= -3$

  2. slope $= 0$, $y$-intercept $= -3$

  3. slope $= 1$, $y$-intercept $= -3$

  4. slope $= 1$, $y$-intercept $= 3$

Show answer
Correct Option: C
Explanation:
The slope intercept form of the line is $y=mx+c$ where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $x-y=3$ in slope intercept form:
$x-y=3$
$\Rightarrow -y=-x+3$
$\Rightarrow y=x-3$
Hence, the slope of the line $x-y=3$ is $m=1$ and the $y$-intercept is $-3$.

A line in the $xy$-plane passes through the origin and has a slope of $\dfrac{1}{7}$. Which of the following points lies on the line?

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\left(0, 7\right)$

  2. $\left(1, 7\right)$

  3. $\left(7, 7\right)$

  4. $\left(14, 2\right)$

Show answer
Correct Option: D
Explanation:

If any straight line passes through origin, then it must of the form $y = mx$.


Now if the slope is $\dfrac{1}{7}$, then line will be $y = \dfrac{1}{7}x \ $ or $ \ 7y -x = 0$

We can see out of all the points only point $(14,2)$ satisfies the equation of the line. Hence Only $(14,2)$ lies on the line.

Correct option is $D$

Find an equation of the line through the points $(-3,5)$ and $(9,10)$ and write it in standard form $Ax+By=C$, with $A>0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $6x-10y=-75$

  2. $5x-12y=-75$

  3. $4x-11y=-65$

  4. $x-6y=-15$

Show answer
Correct Option: B
Explanation:
Given points are $(-3,5)$ and $(9,10)$
The slope of the line is given by:
$m =\dfrac{ (10-5)}{[9-(-3)] }= \dfrac {5}{12}$
The equation becomes: 
$y - 5 = \left (\dfrac {5}{12}\right) [x-(-3)]$ 
$y-5=\left (\dfrac {5}{12}\right)(x+3)$ 
Solve it and get the equation- 
$12y-60=5x+15$ 
$5x-12y=-75$