Lesson 7.1
Checking Solutions
A "System" is just a fancy word for "two or more equations at the same time." Before we learn to solve them, we need to learn how to check if an answer is correct.
Introduction
In the past, a solution was just a number (). Now, a solution is a location () where two lines cross.
Past Knowledge
Lesson 4.8 (Evaluating Functions). You need to be able to plug inputs into equations and simplify.
Today's Goal
Determine if a point check verifies BOTH equations.
Future Success
Checking your work is the superpower of A+ students. It guarantees you never hand in a wrong answer.
Key Concepts
The "Double Check" Rule
To be a solution to the system, the point must make BOTH equations true.
Must be TRUE
Must ALSO be TRUE
If it fails EITHER one, it is NOT a solution.
Worked Examples
Example 1: It Works!
BasicIs a solution to the system?
Equation 1:
Plug in :
(TRUE)
Equation 2:
Plug in :
(TRUE)
Conclusion: Yes.
Since it worked for BOTH, it is a solution.
Example 2: The Half-Truth
BasicIs a solution to the system?
Equation 1:
Plug in :
(TRUE)
Equation 2:
Plug in :
FALSE
Conclusion: No.
It worked for the first one, but failed the second. The point is on one line, but not the intersection.
Example 3: Visual Check
VisualVerify visually if is the solution.
The red dot is exactly where the blue and purple lines meet.
Conclusion: Yes.
The graphical definition of a solution is the Intersection Point.
Common Pitfalls
Stop at One
Students often check the first equation, see that it works, and yell "Yes!". You are only 50% done. You MUST check the second equation.
Mixing x and y
In , the 2 is and the 5 is . Don't plug 2 into ! Label your point if it helps.
Real-Life Applications
GPS Triangulation:
- One satellite tells you that you are 500 miles away (a circle of possibilities).
- A second satellite gives another circle.
- Where they intersect is your exact location. Your phone solves systems of equations constantly to figure out where you are on Google Maps.
Practice Quiz
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