Systems of Equations
Systems of Linear Equations: Substitution and Elimination
Two equations in two unknowns, the two elementary methods, and the geometric meaning of a solution as an intersection.
What this page covers
- Recognise a system as linear
- Solve by substitution
- Solve by elimination
- Interpret the solution as the intersection of two lines
What a system is
A system is several equations that must hold at the same time. A solution is a set of values satisfying every one of them, not just some.
Every variable appears only to the first power. The source gives the test negatively: linear because there are no x2, xy, y2 or √x terms.
Unknowns on the left, constants on the right, like terms aligned in columns.
Rearranging — the source's example
The rearrangement is not cosmetic. Elimination depends on like terms sitting in the same column, and the augmented-matrix methods of Week 11 depend on it absolutely.
Substitution
- Solve one equation for one variable, choosing whichever is easiest to isolate.
- Substitute that expression into the other equation, which then has only one unknown.
- Solve the resulting single equation.
- Back-substitute to find the other variable.
- Check in both original equations.
Worked example — the source's case
Solve 2x + 3y = 2 and 4x - y = 1.
In (1): 1014 + 97 = 57 + 97 = 2 ✓. In (2): 2014 - 37 = 107 - 37 = 1 ✓.
A cleaner case — the source's second
Solve 3x + 2y = 5 and x + y = 2.
The source draws the two lines meeting at (1, 1), which is what the solution means geometrically.
Substitution also handles non-linear systems. The source solves y2 + x = 0 with 2x + 3y = -2 by isolating x = -1 - 32y and substituting, giving 2y2 - 3y - 2 = 0, hence y = 2 or y = -12. Two solutions, because a line can cut a parabola twice.
Elimination
Multiply the equations so that one variable has matching coefficients, then add or subtract to remove it.
- Choose a variable to eliminate.
- Multiply each equation by whatever makes that variable's coefficients equal in size.
- Add or subtract to eliminate it.
- Solve for the remaining variable.
- Back-substitute and check.
Worked example — the source's case
Solve 3x + 2y = 5 and x + y = 2.
Subtract every term, including the constants. Writing the second equation with all its signs reversed on a separate line before adding is slower and far safer than subtracting mentally.
A case needing both equations multiplied
Solve 2x - 3y = 4 and 3x + 2y = 19.
In the second original equation: 3(5) + 2(2) = 15 + 4 = 19 ✓. Checking in the equation not used for back-substitution is the meaningful test.
Which method to choose
| Situation | Better method | Why |
|---|---|---|
| A variable has coefficient 1 | Substitution | Isolating it introduces no fractions |
| All coefficients awkward | Elimination | Multiplying is easier than dividing |
| The system is non-linear | Substitution | Elimination assumes linearity |
| Three or more equations | Elimination | It generalises to Gaussian elimination |
| The system will be solved by computer | Elimination | It is the basis of the matrix algorithms |
Elimination is the more important of the two in the long run. Everything in Week 11 — augmented matrices, row operations, row-echelon form — is elimination made systematic.
The geometric picture
Each linear equation in two unknowns is a line. Solving the system means finding where the lines meet.
| Lines | Solutions | Name |
|---|---|---|
| Cross at one point | Exactly one | Consistent, independent |
| Parallel and distinct | None | Inconsistent |
| Identical | Infinitely many | Dependent |
The source draws all three cases. Two distinct lines in a plane either meet once or not at all; there is no way for them to meet exactly twice. The three cases are developed in Consistent, Inconsistent and Dependent Systems.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Not rearranging to standard form | Align like terms first | Elimination depends on the alignment |
| Subtracting only some terms | Every term, both sides | Write the negated equation on its own line |
| Multiplying only one side of an equation | Multiply both | Otherwise the equation changes |
| Checking in one equation only | Check in both | One equation passes for many wrong pairs |
| Substituting back into the rearranged equation | Use an original | An early error would be reproduced |
| Stopping after finding one variable | Find both | A solution is an ordered pair |
Frequently asked questions
Which method is better?
Substitution when one variable already has a coefficient of 1. Elimination when the coefficients are awkward, or when there are three or more equations — it is what Gaussian elimination generalises.
Why put the system in standard form first?
Because elimination lines up like terms in columns, and the constants must be on the right. The source rearranges every system before solving.
What does the solution mean geometrically?
The point where the two lines cross. The source draws the graphs alongside the algebra for exactly this reason.
Do systems have to be linear?
No, but non-linear systems need more care. The source works y2 + x = 0 with 2x + 3y = -2 by substitution, which produces a quadratic in y and two solutions.
Source. Handwritten teaching notes, Week 10, pages 7-10, with the typed supplementary notes on simultaneous equations.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.