Question

find the equation of a line passing through the point of intersection of the lines x+y+5=0 and having x- intercept 3.

Answer 1

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

Using the 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.

7x−3y−19=0→(1)

3x+2y+5=0→(2)

Note:−3y×2=−6y and 2y×3=6y

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

(1)×2:14x−6y−38=0→(3)

(2)×3:9x+6y+15=0→(4)

(3)+(4) term by term

⇒23x+0y−23=0← equation in one variable

⇒23x=23⇒x=1←value for x

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

Substitute x=1 in (2)

⇒(3×1)+2y+5=0

⇒8+2y=0⇒2y=−8⇒y=−4←value for y

As a check

Substitute these values into ( 1 )

(7×1)−(3×−4)−19=7+12−19=0→ true

⇒(1,−4) is the point of intersection

Reminder ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣∣22 parallel lines have equal slopes22∣∣∣−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

y=2x+1 is in slope-intercept form

⇒slope =m=2

Expressing the required equation in  point-slope form

y−y1=m(x−x1) with m=2 and (x1,y1)=(1,−4)

⇒y+4=2(x−1)← in point-slope form

distribute and simplify.

y+4=2x−2

⇒y=2x−6← in slope-intercept form

Answer 2

you take an equation and put in