# 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?

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

#"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.

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

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

#"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)xx2: 14x-6y-38=0to(3)#

#(2)xx3: 9x+6y+15=0to(4)#

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

#rArr23x+0y-23=0larrcolor(blue)" equation in one variable"#

#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)#

#rArr(3xx1)+2y+5=0#

#rArr8+2y=0rArr2y=-8rArry=-4larrcolor(blue)"value for y"#

#color(blue)"As a check"# Substitute these values into ( 1 )

#(7xx1)-(3xx-4)-19=7+12-19=0to" true"#

#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"#

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

If the straight be parallel to the straight line

then

Inserting the value of k in [1]

This is the equation of the required straight line.

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Contd.

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We will use the following well-known

**Result :** The eqn. of a line

intersection of the lines

the form

Result :

Describe your changes (optional) 200