Without graphing or solving, how many solutions?
Classify: Consistent/Inconsistent, Dependent/Independent.
How to start the problem:
Put each equation into 𝑦 = 𝑚𝑥 + 𝑏 form
(Slope-Intercept form, get the 𝑦 by itself.)
EXAMPLE: Consider these two equations
2𝑥 + 𝑦 = 1
3𝑥 – 5𝑦 = −5
Add −2𝑥 on each side:
Add −3𝑥 on each side:
𝑦 = −2𝑥 + 1
−5𝑦 = −3𝑥 – 5
Divide each term by −5:
3
𝑦 = − 𝑥+1
5
How to decide on the answers:
Look at the SLOPES first.
(the m in y = mx + b)
If the slopes are the same, then look at the Y-INTERCEPTS.
(the b in y = mx + b)
If the slopes are different, it’s like
the two lines intersecting in one
point. Here are the answers:
 “There is exactly ONE
solution.”
If the y-intercepts are different, it’s like the two parallel lines
situation in graphing. Here are the answers:
 “There are NO SOLUTIONS”.
 “The system is INCONSISTENT”
 And we say nothing about dependent or independent.
 “The system is
CONSISTENT.”
 “The equations are
INDEPENDENT.”
But if the y-intercepts are the same, that is, the two equations
are really the same equation. If the two lines were graphed, it
would really be the same line, one on top of the other.
 “There are INFINITE NUMBER of solutions.”
 “The system is CONSISTENT”.
 “The equations are DEPENDENT”.
EXAMPLE, continued:
3
The slopes are −2 and − 5
In our example, the slopes are
the same, so our answers are:
“There is exactly one solution;
the system is consistent;
the equations are
independent.”
Note: The y-intercepts aren’t
even in the game here. As
soon as we found the slopes
were different, that was all
we needed.
4NatureOfSolutions.docx
4/26/2014 1:06 PM - D.R.S.