Find equation of the straight line satisfying some conditions

Question

Can some one please help me to solve the following question?

Find equation of the straight line that goes through $(2 , -5 )$ and:

$1$. has slope $-3$

$2$. Is parallel to the $x-axis$

$3$. Is parallel to the $y-axis$

$4$. Is parallel to the line $2x - 4y = 3$

Answer 1

For general equation we have $(y-y_1)=m(x-x_1)$ we have for 1 $y_1=2,x_1=-5,m=-3$ for 2 . Remember any line parallel to y-axis has slope $m=\frac{1}{0}$ . Then for 3 any line parallel to x-axis has slope $m=0$ and for 4 parallel lines have slope equal here the slope of given line is $\frac{-a}{b}=\frac{1}{2}$ now plug in the values in above general equation and your work is done.hope it helps you.

Answer 2

hint: the Ansatz is $y=mx+n$ with $m=-3$ (the slope is given) plugging the coordinates ion the equation we get $$-5=-6+n$$ thus $n=1$ for d) we have $$y=\frac{1}{2}x-\frac{3}{4}$$ the parallel straight line is given by $$y=\frac{1}{2}x+n$$ inserting $x=2$ and $y=-5$ we obtain $n=-6$

Answer 3

A line with a slope $m$ passing through points $(a,b)$ is given by $y - b = m(x-a)$

For part b, a line parallel to x-axis has a slope of 0

Answer 4

Ok, let me explain the general equation of the line considering the fact that the other answers already solve the question but you still seem to have problems with understanding them.

Let's say you have a line. The equation of the line comes from the equation of the slope. Let the slope of the line be $$ m $$

Now, by definition, given two points $$ (a, b) \thinspace and \thinspace (c, d) $$ the equation of the slope is: $$\begin{aligned} m = \dfrac{d - b}{c - a} \end{aligned}$$ Multiplying by $$ c - a $$ we get $$m * (c - a) = (d - b) $$ By substituting m in the above equation we get $$\begin{aligned} d - b = \dfrac{d - b}{c - a} * (c - a) \end{aligned}$$ This equation is valid for any two points that are on the line. Therefore if you know the slope of a line and another point that lies on it, say $$ (k, l)$$ it's easy to determine whether another point $$(x,y)$$ lies on the given line. Thus $$\begin{aligned} y - l = m * (x - k) \end{aligned}$$

Also, note that two parallel lines have the same slope, and the slopes $$m_1,m_2$$ of two perpendicular lines satisfy the equation $$m_1 * m_2 = -1$$

Hope this helps!

Answer 5

The general equation of a straight line in the plane is the linear equation:

$$ ax+by+c=0 $$

If the point $P=(2,-5)$ is on the line than its coordinate satisfies the equation so that we have:

$$ 2a-5b+c=0 \Rightarrow c=5b-2a $$

so, all lines through $P$ ( the bundle of stright lines through the point), have equation: $$ ax+by+5b-2a=0 $$

Now:

if $a=0$ the line becomes $by+5b=0$ i.e. $y=-5$ that is a straight line wich points have fixed $y$ for all possible $x$, so it is a line parallel to the $x$ axis;

if $b=0$ the line becomes $ax-2b=0$ i.e. $x=2$ that is a straight line wich points have fixed $x$ for all possible $y$, so it is a line parallel to the $y$ axis;

if $a \ne 0$ and $b \ne 0$ the line becomes: $$y=-\frac{a}{b}x+\frac{2a}{b}-5 $$ where $-\frac{a}{b}$ is the slope of the straight line.

So, if we want a slope $-3$ we have $-\frac{a}{b}=-3 \Rightarrow \frac{a}{b}=3$ and the equation of the line becomes: $y=-3x+1$.

For the question d), note that the given line has equation: $y=\frac{1}{2}x-\frac{3}{4}$, so has slope $\frac{1}{2}$, a parallel line have the same slope, so substitute this slope and you find the searched parallel line.