- Idea
- 2 equations in 2 unknowns (variables)
- Can be solved for both unknowns
- An example of a system of equations in 2 unknowns:
| 2x + 3y = 7 |
(1) |
| 3x + 2y = 8 |
(2) |
- Observe that for the above example:
- (2, 1) satifies both equations.
- Plug x = 2 and y = 1 into equations (1) and (2), and both
equations hold true.
- (5, -1) does not satisfy both equations.
- Plug x = 5 and y = -1 into equations (1) and (2), and
equation (1) holds true but equation (2) is false.
- Conclusion:
- (2, 1) is a solution to the example system of equations.
- (5, -1) is not a solution to the example system of equations.
- Notice that equations (1) and (2) are each equations of lines in standard form.
- Three methods (techniques) of solution studied in this section
- Substitution
- Elimination
- Graphical
- Solution Types
| Consistent |
1 solution |
Lines intesect in 1 point |
| Inconsistent |
No solutions |
Lines are parallel |
| Dependent |
Infinitely many solutions |
Lines are the same |
- Substitution Method
- Steps to solution...
- Solve one of the equations for one of the variables.
- Substitute in the other equation: the expression for the
variable found in step 1.
- Solve the resulting equation (that is an equation in one variable after step 2).
- Substitute the value of the variable found in step 3 into either of the original
equations and solve for the value of the other variable.
- You now have values for the 2 variables (i.e., you have the solution).
Check that the solution satisfies both equations.
- Elimination Method--Also known as the Addition Method
- Idea: Multiply one, or both, equations by a number, or numbers, so that
one of the variables dissappears when you add the equations together.
Then you can solve the resulting equation for a value of a variable and
substitute that value back into one of the original equations to find
the other variable.
- Steps to solution...
- Write both given equations in standard form (best to get rid of fractions in
this step!).
- Multiply one, or both, equations by a number, or numbers, so that
one of the variables dissappears when you add the equations together.
- Add the equations together. That means, add the left-hand-sides to get the
left-hand-side of the resulting equation and add the right-hand-sides to get the
right-hand-side of the resulting equation.
- You now have an equation in 1 variable--solve it for the value of that variable.
- Substitute the value of the variable found in step 4 into either of the original
equations and solve for the value of the other variable.
- You now have values for the 2 variables (i.e., you have the solution).
Check that the solution satisfies both equations.
- Graphical Method
- Steps to solution...
- Graph the 2 equations.
- The solution is the intersection of the 2 lines.
- The lines may intersect in 1 point meaning they are consistent (i.e., the
system of equations has 1 solution).
- The lines may be the same (colinear) meaning they are dependent (i.e., the
system of equations has infinitely many solutions).
- The lines may be parallel meaning they are inconsistent (i.e., the
system of equations has no solutions).
- Look at the intesection and write down the solution.
- Check that the solution satisfies both equations.
- The problem with this method is that it is often difficult to accurately write down
the solution by looking at the point of intersection. For example, it is hard to
look at a graph and tell that a point is (1.625, 3.875).
- The usefulness of this method is that you can look at the intersection of two lines
and tell if the solution point you found, by the substitution or elimination method,
is close to correct.
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