The solution(s) to a system of linear equations are all the point(s) where the lines intersect. To solve a system by graphing, we graph the lines and see where they meet up. Are they grabbing a couple of lattes at Starbucks, or is this more of a public park rendezvous?
Of course, if we want to get the right answer, we need to draw the graphs carefully. Now is not the time to challenge yourself by attempting to draw the graph with your non-dominant hand.
Sample Problem
Solve the following system of equations by graphing:
If we graph these equations, we get this picture:
We see that the lines intersect at (1, 1). Let's make sure this is actually a solution to the system of equations. First, we check the values in the first equation:
y = x
1 = 1
Yup, the values x = 1 and y = 1 are definitely solutions to this equation.
Now, let's check the values in the second equation:
y = 2 – x
1 = 2 – 1
When x = 1 and y = 1, the left-hand side of this equation is 1 and the right-hand side is 2 – 1, which is also 1. It's true; you can use your fingers to check our math.
The point (1, 1) is a solution to each equation in the system. In other words, the point (1, 1) is on both lines. This means (1, 1) is a solution to the system of equations. It is also a common final soccer game score.
Sample Problem
Solve the following system of linear equations:
4x + 3y = 24
y = (-4/3)x + 8
First we graph the equation 4x + 3y = 24 by looking at the intercepts:
Now we graph the equation y = (-4/3)x + 8, which also has a y-intercept of 8 and a slope of -4/3:
Somebody's a copycat. The second line we graphed landed right on top of the first one. The first line wasn't ready for it, either. It jumped a mile. It was great.
What the presence of overlapping lines means is that the two lines are actually the same and intersect at every point along the line. These two are always shaking hands and slow dancing. If they're being honest, it's getting a little old. What, they can't groove to something up-tempo every once in a while?
All points on the line y = (-4/3)x + 8 are solutions to this system of equations.
We could also say all points on the line 4x + 3y = 24 are solutions, since that's a different way to write the same line. Two ways to write the same line...a screenwriter with writer's block would be envious.