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4.1 Systems of Linear Equations
A. Introduction
Suppose we have two lines.
We have three possibilities:
the lines intersect the lines are parallel the lines overlap (coincide)
Thus, if we have a system of linear equations ; that is, “two line equations in ^ and ^ ”, then...
If the
- lines intersect: we have one ordered pair that lies on both lines
- lines are parallel: no ordered pairs lie on both lines
- lines coincide: infinitely many ordered pairs lie on both lines
Ordered pairs ^ ^ ^ ^ that work in both equations are called solutions to the system of equations. They represent the intersection points of the two lines. Thus a system has one solution , no solutions , or infinitely many solutions.
B. Checking Solutions to Systems
If an ordered pair is a solution, it must work in both equations. We plug the trial point into each system. It is a solution only if it works in both.
Example: Check to see if ^ ^ ^ ^ is a solution to the system
Solution
Plug ^ ^ ^ ^ into each equation and see if the equation is true. ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ X
Doesn’t work in both!
Ans not a solution
C. Solving a System By Substitution
- Pick one of the two equations (your choice).
- Solve the equation for one of the two variables (your choice).
- Substitute into the other equation with that variable.
- After solving, the answer is one of the coordinates.
- Substitute back into any equation to get the other variable answer.
If we had picked equation 1 and solved for ^ :
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
Replacing ^ with
in equation 2:
^
Then ^ ^
, so^
^ ^ ^ ^ ^
^ ^ ^ ^
Thus we get ^ ^ ^ ^ as before.
D. Comments on Solving Systems
- If when solving a system, you get a false statement (like ^ ), the system has no solutions.
- If when solving a system, you get a true statement (like ^ ^ ^ ), the system has an infinite number of solutions (namely all points on the common line).
The above situations happen when the variables “disappear” during the solution process.
E. Solving a System By Elimination
Another way to solve a system (sometimes easier) is to eliminate one of the variables.
- Get coefficients opposite in sign of the variable you want to eliminate. Do this by multiplying each equation by a number.
- Add equations to eliminate the variable.
- Solve the equation to get one coordinate.
- Substitute back into one of the equations to get the other coordinate.
Example 1: Solve the system
(^) for^
(^) by elimination.
Solution Notice that by adding both equations the variable ^ disappears: ^ ^ ^ ^ ^.
Then ^ ^ ^.
Plugging into the second equation, say, we get ^ ^ ^ ^ ^ , so ^ ^ .
Thus, we have ^ ^ ^ ^ ^ ^ ^
Ans ^ ^
Example 3: Solve the system
^ ^ ^
^ ^ ^ ^ ^ ^
for ^ ^ ^ ^ by elimination.
Solution
We decide to eliminate ^ (arbitrary choice). To cancel, we must multiply by constants...
Multiply equation 1 by and equation 2 by ^ ^ : (^)
Now add the equations: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^.
Plug ^ ^ ^ ^ into equation 2 (your choice):
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
Thus ^ ^ ^ ^ ^ ^ ^ ^ ^ ^.
Ans ^ ^ ^ ^
Example 4: Solve the system
^ ^ ^ ^
for^
(^) by elimination.
Solution
We decide to eliminate ^ (arbitrary choice). To cancel, we must multiply by constants...
Multiply equation 1 by ^ : (^)
Now add the equations: ^
The variables disappeared and we got a true statement!
Thus we have infinitely many solutions.
Ans infinitely many solutions! all points on the line ^ ^ ^ ^
F. Closing Comments
- If a system of equations has no solution, we sometimes say that the system is inconsistent.
- If a system of equations has infinitely many solutions, we say that the equations are dependent.
- Another solution method is called solution by graphing. Here each line is graphed, and the intersection point is determined by picture. However, this is usually difficult to get an accurate answer.
