Ex 1: Find the Equation of a Line Parallel to a Given Line Passing Through a Given Point – Howtoshtab – how to, lifehacks, tips and tricks

– WE WANT TO DETERMINE THE EQUATION OF THE LINE THAT IS PARALLEL TO Y = -5X + 4 AND PASSES THROUGH THE POINT WITH COORDINATES (-1,4). THE MAIN THING TO REMEMBER HERE IS THAT IF LINES ARE PARALLEL, AS WE SEE HERE ON THE RIGHT, THE SLOPES ARE EQUAL TO EACH OTHER. SO IF WE WANT A LINE THAT IS PARALLEL TO Y = -5X + 4, THE SLOPE OF THE NEW LINE MUST BE -5 AND IT PASSES THROUGH THIS POINT. SO NOW WE NEED TO FIND THE EQUATION OF A LINE WITH A SLOPE OF -5 THAT CONTAINS THE POINT (-1,4).

NOW, THERE ARE TWO MAIN WAYS OF DETERMINING THE EQUATION OF A LINE GIVEN THE SLOPE AND A POINT. WE CAN USE JUST SLOPE INTERCEPT FORM, WHICH IS Y = MX + B OR WE COULD USE POINT SLOPE FORM, WHICH IS Y – Y1 = M(X – X SUB 1). I’M GOING TO GO AND SHOW THIS BOTH WAYS. SO IF WE USE SLOPED INTERCEPT FORM WE CAN SUBSTITUTE -5 FOR M. SO WE’D HAVE Y = -5X + B, BUT WE STILL NEED TO DETERMINE THE Y INTERCEPT. AND THIS Y COORDINATE IS NOT THE Y INTERCEPT BECAUSE X IS NOT = 0, BUT THIS POINT IS ON THE LINE SO WE CAN SUBSTITUTE THESE VALUES INTO THE EQUATION AND THEN SOLVE FOR B. SO WE’LL SUBSTITUTE -1 FOR X AND POSITIVE 4 FOR Y TO DETERMINE B. SO WE’D HAVE 4 = -5 x -1 + B. SO WE HAVE 4 = 5 + B. SO NOW WE’LL SUBTRACT 5 ON BOTH SIDES TO SOLVE FOR B. SO WE HAVE B = -1. SO NOW WE CAN SUBSTITUTE -1 FOR B INTO THIS EQUATION HERE AND WE HAVE THE EQUATION IN SLOPE INTERCEPT FORM.

WE WOULD HAVE Y = -5X AND THEN + -1 IS EQUIVALENT TO – +1. SO THIS WOULD BE OUR EQUATION IN SLOPE INTERCEPT FORM. AND I’LL SEE IF WE GET THE SAME EQUATION IF WE USE POINT SLOPE FORM INSTEAD. REMEMBER IN POINT SLOPE FORM M IS STILL THE SLOPE, BUT X SUB 1, Y SUB 1 ARE THE CORNERS OF ANY POINT ON THE LINE. SO WE’LL USE -1 FOR X SUB 1 AND 4 FOR Y SUB 1. SO WE WOULD HAVE Y – 4 = -5(X). AND THEN HERE WE’D HAVE – -1, WHICH IS EQUIVALENT TO 1. AND NOW TO WRITE THIS IN SLOPE INTERCEPT FORM WE’LL SOLVE FOR Y SO WE’LL DISTRIBUTE HERE. SO WE’D HAVE Y – 4 = -5X – 5 AND THEN WE ADD 4 TO BOTH SIDES, THIS WOULD BE ZERO. AND YOU CAN SEE WE’RE GOING TO HAVE THE SAME EQUATION. WE’LL HAVE Y = -5X AND THEN THIS IS -1. SO AS YOU CAN SEE WHICHEVER METHOD YOU USE THE RESULT IS THE SAME. IT JUST DEPENDS ON WHAT METHOD YOU WANT TO USE. LET’S GO AHEAD AND QUICKLY VERIFY THAT THIS LINE IS PARALLEL TO THE GIVEN LINE AND CONTAINS THE POINT (-1,4). RIGHT AWAY WE CAN SEE THEY HAVE THE SAME SLOPE SO THEY ARE PARALLEL.

THE GIVEN EQUATION PRODUCED THE RED LINE AND THE EQUATION THAT WE FOUND PRODUCED THE BLUE LINE. NOTICE HOW THE BLUE LINE DOES CONTAIN THE POINT (-1,4) RIGHT HERE. HAS THE SAME SLOPE AS THE RED LINE, WHICH WAS -5 AND IT DOES HAVE A Y INTERCEPT OF -1. SO THIS VERIFIES OUR WORK IS CORRECT..         