This week we learned about how to use substitution to find the solution of two linear relations. The substitution method is essentially where you take one equasion and input it into the second one. The solution is the point where the two relations would cross if they were graphed. It is possible to have lines with one solution (where the two lines cross at one point on the graph), no solutions (if the lines are parallel and the slopes are the same) and infinite solutions (if the two lines are on top of one another and have identical equations). To find the solution using the substitution method you’d need to follow the following steps. I’ll use the equations (x+4y=-3) and (3x-7y=29) as an example.
- First you would need to isolate a variable. Select a variable from either equation that seems the easiest to work with. The variable that seems the easiest to work with in these equations is the x from the first equasion (x+4y=-3) as it does not have a base associated with it. After you have chosen your variable, you can isolate it by subtracting 4y from both sides of the equal sign. The rearranged equation should look like this: (x=-4y-3).
- Next you can take (-4y-3) and substitute it into the second equation (3x-7y=29). Since x in the first equation is equal to (-4y-3) you would replace the x in the second equation with that same answer. Now the second equation should look like this: (3(-4y-3)-7y=29).
- After that you can solve the equation. Take the 3 and use the distributive law first to get rid of the brackets. The equation then becomes (-12y-9-7y=29). Then, simplify the (-12y) and the (-7y) so that the equation becomes (-19y-9=29). After you can add 9 to both sides of the equal sign so the equation becomes (-19y=38). Finally divide both sides by (-19) to get (y=-2). Now you have the y coordinate of the solution. All we need is to find the x.
- To do that, substitute the y coordinate (-2) back into the rearranged version of the first equation (x=-4y-3) so that it becomes (x=-4(-2)-3). Solve this equation to find x. (-4(-2)) is equal to (8) and (8-3) is equal to (5). Therefore, (x=5).
Now you know that the solution of these two equations and you can write it as an ordered pair (5, -2). You can verify this by using a graphing calculator or by inputting the two numbers back into one or both of the original equations, (x+4y=-3) and (3x-7y=29). If we use the first one, it would be (5-8=-3) so we know it is accurate. You can also see on the graph showed here that the two lines cross at (5, -2).