1. Solving Systems of Equations
259 Lecture 12 Spring 2017
Solving Systems of Equations

2. A System of Equations
Consider the system of two equations in two unknowns:
x + 2y = 1 (1)
3x + 4y = -1 (2)
Recall from algebra class that two ways to solve a problem like this are substitution and elimination.

3. Example 1 (Substitution)
Solve (1) or (2) for one variable in terms of the other and substitute for that variable in the other equation.
x + 2y = 1 (1)
3x + 4y = -1 (2)
Solving (1) for x yields
x = 1 – 2y (3)

4. Example 1 (cont.) Using (3) we can substitute 1 – 2y for x in (2):
3(1 – 2y) + 4y = -1
3 – 6y + 4y = -1
-2y = -4
y = 2 (4)
(4) in (3) implies x = 1 – 2(2) = -3.
Check (x, y) = (-3, 2) solves (1), (2).

5. Example 2 (Elimination)
x + 2y = 1 (1)
3x + 4y = -1 (2)
Multiply both sides of (1) or (2) by a non-zero constant to get coefficients in front of one variable that can be added to get zero.
Then add to eliminate that variable and solve for the remaining variable.

6. Example 2 (cont.)
Multiplying (1) by -2 on both sides, we get an equivalent system;
-2x - 4y = -2 (3)
3x + 4y = -1 (2)
Adding (3) to (2) gives an equation involving only x which can be solved for x.
x = (4)

7. Example 2 (cont.)
Substituting x = -3 from (4) back into either (1) or (2) (why these?), we can solve for y.
In this case, we choose (1):
x + 2y = 1
-3 + 2y = 1
2y = 4
y = 2
Again, we find (x, y) = (-3, 2) and would need to check it works in (1),(2)!

8. Solving a System of Equations with Technology
How could we use technology to help us solve system (1), (2).
One way might be to use a calculator with this capability.
How about Mathematica or Excel?

9. Example 3: Solving a System of Equations in Excel
We can use the Solver to solve a system of equations!
Let’s try with (1), (2).
To do so, we need to choose one target cell (i.e. objective cell) and specify appropriate constraints.
Think of the system as
a*x + b*y = e
c*x + d*y = f,
with a=1, b=2, c=3, d=4, e=1, and f=-1.
Then the objective can be to set ax+by = 1, subject to the constraints e = 1 and f = -1.

10. Example 3: Solving a System of Equations in Excel

11. Example 4: Solving a System of Equations in Mathematica
For Mathematica, we use the Solve command:
Solve[{x+2y==1,
3x+4y==-1},{x,y}]

12. Example 4: Solving a System of Equations in Mathematica
We can also solve a system like this in general:
To solve the system
a*x + b*y = e
c*x + d*y = f,
use the command:
Solve[{a*x+b*y==e,
c*x+d*y==f},{x,y}]

13. Example 4: Solving a System of Equations in Mathematica
We find that for the system
a*x + b*y = e
c*x + d*y = f,
the solution is:
x = (bf – de)/(ad – bc)
y = (af – ce)/(ad – bc)
Try with coefficients and right-hand side from (1), (2) to see if we get the same solution!
For what choices of a, b, c, d, e and f, do we get a solution to this system?

14. Systems of Equations in General
We can generalize the problem discussed above to a system of n equations in n unknowns!
For example, here is a system of three equations in three unknowns:
ax + by + cz = j
dx + ey + fz = k
gx + hy + iz = l
How could we solve this?
By hand?
Calculator?
With Mathematica?
With Excel?
Other?
When are we guaranteed a solution?

15. Example 5: Mathematica Solution for 3 x 3 Case

16. Example 5 (cont.)

17. References
The idea of how to use the Solver comes from a web supplement for Finite Mathematics (7th ed) by Margaret Lial et al. found at this link: