2. Systems of Linear Equations
We have been studying systems, and we know that when graphing,
we can encounter three possibilities:
1. Intersecting lines with one solution (the point where they
intersect)
2. Parallel lines with no solutions (they never intersect)
3. Coincidental lines with infinitely many solutions (they always
intersect)
3. We can find these
solutions by graphing.
However, sometimes we just want to know if a point is a solution. While
graphing is an option, it is time-consuming in many cases. How else
could we verify whether or not a point is a solution?
5. Let’s take a look at an
example.
Write this one down in your notes!
6. Is (0, -2) a solution of the system
2𝑥 − 3𝑦 = 6
𝑦 = 7𝑥 − 2
?
– First, notice that 0 is x and -2 is y.
– Then, substitute these values into the first equation to see if we get a true
statement: 2 0 − 3 −2 = 6
– When we simplify, we see 0 + 6 = 6, which is a true statement.
– Next, let’s substitute these values into the second equation to see if we get a
true statement: −2 = 7 0 − 2
– When we simplify, we see −2 = 0 − 2, which is a true statement.
Since we have true statements from each equation, we know that (0, -2) is a
solution of the system.
7. Is (1, -1) a solution of the system
𝑛 = 2𝑚 − 3
3𝑚 − 𝑛 = 2
?
– First, notice that 1 is m and -1 is n. Coordinates are always given to us in
alphabetical order. m is like x in this case, and n is like y.
– Then, substitute these values into the first equation to see if we get a true
statement: −1 = 2 1 − 3
– When we simplify, we see −1 = 2 − 3, which is a true statement.
– Next, let’s substitute these values into the second equation to see if we get a true
statement: 3 1 − −1 = 2
– When we simplify, we see 3 + 1 = 2, which is a false statement.
Since we don’t have true statements from each equation, we know that (1, -1) is NOT
a solution of the system.
