How to find the equation of the line which passes through the point of intersection of the lines 7x − 3y − 19 = 0 and 3x + 2y + 5 = 0, give that the line is parallel to the line with the equation y = 2x + 1?

Question

How to find the equation of the line which passes through the point of intersection of the lines 7x − 3y − 19 = 0 and 3x + 2y + 5 = 0, give that the line is parallel to the line with the equation y = 2x + 1?

3 Answers

Impact of this question

98116 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-27T09:58:58

Contd.

Explanation:

We will use the following well-known

Result : The eqn. of a line $l$ $l$ passing through the pt. of

intersection of the lines $l_{1} : a_{1} + b_{1} + c_{1} = 0 and l_{2} : a_{2} x + b_{2} y + c_{2} = 0$ $l_1 : a_1+b_1+c_1=0 and l_2 : a_2x+b_2y+c_2=0$ is of

the form

Result :

Answer 2 · 2017-03-27T10:00:35

Let equation of any straight line passing through the point of intersection of two given straight line be

$k (7 x - 3 y - 19) + (3 x + 2 y + 5) = 0$ $k(7x-3y-19)+(3x+2y+5)=0$

$\Rightarrow (7 k + 3) x + (2 - 3 k) y + (5 - 19 k) = 0 . . [1]$ $=>(7k+3)x+(2-3k)y+(5-19k)=0.. [1]$

If the straight be parallel to the straight line $2 x - y + 1 = 0$ $2x-y+1=0$

then

$\frac{7 k + 3}{2} = \frac{2 - 3 k}{- 1}$ $(7k+3)/2=(2-3k)/(-1)$

$\Rightarrow 7 k + 3 = - 4 + 6 k$ $=>7k+3=-4+6k$

$\Rightarrow k = - 7$ $=>k=-7$

Inserting the value of k in [1]

$(7 \cdot (- 7) + 3) x + (2 - 3 \cdot (- 7)) y + (5 - 19 \cdot (- 7)) = 0$ $(7*(-7)+3)x+(2-3*(-7))y+(5-19*(-7))=0$

$\Rightarrow - 46 x + 23 y + 138 = 0$ $=>-46x+23y+138=0$

$\Rightarrow 2 x - y - 6 = 0$ $=>2x-y-6=0$

This is the equation of the required straight line.

Answer 3 · 2017-03-27T10:22:48

$y = 2 x - 6$ $y=2x-6$

Explanation:

The first step is to find the point of intersection of the 2 lines.

$Using the elimination method$ $"Using the "color(blue)"elimination method"$

That is we attempt to eliminate the x or y term from the equations leaving us with an equation in 1 variable which we can solve.

Labelling the equations.

$7 x - 3 y - 19 = 0 \to (1)$ $color(red)(7x)color(magenta)(-3y)-19=0to(1)$

$3 x + 2 y + 5 = 0 \to (2)$ $color(red)(3x)color(magenta)(+2y)+5=0to(2)$

$Note: - 3 y \times 2 = - 6 y and 2 y \times 3 = 6 y$ $"Note:" color(magenta)(-3y)xx2=color(magenta)(-6y)" and "color(magenta)(2y)xx3=color(magenta)(6y)$

That is the y terms have the same coefficient but with opposing signs. Hence summing them will result in their elimination.

$(1) \times 2 : 14 x - 6 y - 38 = 0 \to (3)$ $(1)xx2: 14x-6y-38=0to(3)$

$(2) \times 3 : 9 x + 6 y + 15 = 0 \to (4)$ $(2)xx3: 9x+6y+15=0to(4)$

$(3) + (4) term by term$ $(3)+(4)" term by term"$

$\Rightarrow 23 x + 0 y - 23 = 0 \leftarrow equation in one variable$ $rArr23x+0y-23=0larrcolor(blue)" equation in one variable"$

$\Rightarrow 23 x = 23 \Rightarrow x = 1 \leftarrow value for x$ $rArr23x=23rArrx=1larrcolor(blue)"value for x"$

Substitute this value into either of ( 1 ) or ( 2 ) and solve for y

$Substitute x = 1 in (2)$ $"Substitute " x=1" in " (2)$

$\Rightarrow (3 \times 1) + 2 y + 5 = 0$ $rArr(3xx1)+2y+5=0$

$\Rightarrow 8 + 2 y = 0 \Rightarrow 2 y = - 8 \Rightarrow y = - 4 \leftarrow value for y$ $rArr8+2y=0rArr2y=-8rArry=-4larrcolor(blue)"value for y"$

$As a check$ $color(blue)"As a check"$

Substitute these values into ( 1 )

$(7 \times 1) - (3 \times - 4) - 19 = 7 + 12 - 19 = 0 \to true$ $(7xx1)-(3xx-4)-19=7+12-19=0to" true"$

$\Rightarrow (1, - 4) is the point of intersection$ $rArr(1,-4)color(red)" is the point of intersection"$

$color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(" parallel lines have equal slopes")color(white)(2/2)|)))$

$y=2x+1" is in " color(blue)"slope-intercept form"$

$rArr"slope " =m=2$

Expressing the required equation in $color(blue)" point-slope form"$

$y-y_1=m(x-x_1)" with " m=2" and " (x_1,y_1)=(1,-4)$

$rArry+4=2(x-1)larrcolor(red)" in point-slope form"$

distribute and simplify.

$y+4=2x-2$

$rArry=2x-6larrcolor(red)" in slope-intercept form"$

Science

Math

Humanities

... and beyond

How to find the equation of the line which passes through the point of intersection of the lines 7x − 3y − 19 = 0 and 3x + 2y + 5 = 0, give that the line is parallel to the line with the equation y = 2x + 1?

3 Answers

Explanation:

Explanation:

Related questions

Impact of this question