AlgebraAlgebra 1High school

Systems of Equations

Solve two or three linear equations by substitution, elimination or matrices.

A system of two linear equations asks for the point that satisfies both at once, which is where their lines cross. Substitution solves one equation for a variable and plugs it into the other; elimination adds or subtracts multiples of the equations so one variable disappears.

Graphically there are three possibilities: the lines cross once (one solution), they are parallel (no solution), or they are the same line (infinitely many solutions). In elimination these show up as a single value, a false statement such as $0 = 5$ , or a true statement such as $0 = 0$ .

Key terms

Elimination: Add multiples of equations so one variable cancels.

Key ideas

Choose substitution when a variable already has coefficient 1 or $-1$ .
Choose elimination when coefficients line up or can be matched with small multipliers.
Always check the pair in both original equations.
Larger systems are solved the same way, or with matrices and row reduction.

Worked example

Solve $2x + 3y = 12$ and $x - y = 1$ .

From the second equation, $x = y + 1$ .
Substitute: $2(y + 1) + 3y = 12$ , so $5y + 2 = 12$ and $y = 2$ .
Back-substitute: $x = 2 + 1 = 3$ .
Check: $2(3) + 3(2) = 12$ and $3 - 2 = 1$ .

Answer: $(3, 2)$

Common mistakes

Multiplying only one side of an equation when scaling for elimination.
Finding one variable and forgetting the other.
Reporting $0 = 0$ as no solution.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Solve $x + y = 10$ , $x - y = 4$ . What is $x$ ?
Show answer
$x = 7$
2.Solve $3x + 2y = 16$ , $x = 2y$ . What is $y$ ?
Show answer
$y = 2$
3.Solve $4x - y = 9$ , $2x + y = 3$ . What is $x$ ?
Show answer
$x = 2$

Systems of Equations

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

Keep practising

Calculators

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