TSI Math Study Guide
Systems of Equations
On the TSI you should be able to solve a system of two linear equations by substitution, elimination, or graphing, and interpret the solution in a real-world situation.
What the solution means
Each equation is a line. The solution is the point where the lines intersect.
y = −x + 7
Solving by substitution
y = 3x − 4
2x + y = 11
Solving by elimination
2x + y = 11
3x − y = 4
When the variables don’t cancel immediately
Multiply one or both equations first.
2x + 3y = 12
x + y = 5
−2x − 2y = −10now the x terms are opposites y = 2add x + 2 = 5substitute into x + y = 5 x = 3 → (3, 2)
When you multiply an equation, multiply every term on both sides.
Solving by graphing
Graph both lines; the solution is where they cross.
y = x + 1
y = −x + 5(2, 3) so x = 2, y = 3
Graphing shows what a solution means. Substitution or elimination is more precise when the intersection isn’t on clean grid coordinates.
One, none, or infinitely many solutions
One solution
Different slopes — the lines cross exactly once.y = 2x + 1
y = −x + 7
No solution
Same slope, different intercepts — parallel lines. Algebra produces an impossible statement like 0 = 6.y = 3x + 2
y = 3x − 4
Infinitely many
The same line written two ways. Algebra produces an always-true statement like 0 = 0.y = 2x + 4
2y = 4x + 8
Choosing a method
Substitution
A variable is already isolated or has a coefficient of 1.y = 4x − 2
Elimination
The coefficients of one variable are already equal or opposite.3x + 2y = 10
5x − 2y = 14
Graphing
The equations are easy to graph, or you need to interpret the intersection visually.
Before calculating
Take a few seconds to see which method requires the least work.
Writing a system from a word problem
Define the variables
s = student tickets
a = adult tickets
Translate each fact
120 tickets total: s + a = 120
$780 collected: 5s + 8a = 780
The hard part
Usually not solving the equations — it’s translating the information into the correct equations.
Interpreting solutions in context
When a system comes from a real situation, the ordered pair has a specific meaning. If x is adult tickets and y is student tickets and you find (45, 70), the answer isn’t “45 and 70” — it’s 45 adult tickets and 70 student tickets.
If a variable counts people or tickets, a negative answer isn’t reasonable. A solution that doesn’t make sense usually means an error upstream.
TSI strategy: choose the easiest variable to eliminate
Inspect the equations before doing anything.
A variable is isolated
Substitution is probably fastest.
Opposite coefficients, like +4y and −4y
Elimination — just add.
Coefficients can easily be made opposites
Multiply one equation first, then eliminate.
Don’t start calculating the moment you see a system. Choosing the right method turns a long problem into a short one.
A solution must make both equations true. After finding an ordered pair like (4, −2), substitute x = 4 and y = −2 into both originals. A quick check catches sign errors and arithmetic slips before you select an answer.