Practice Systems of linear equations
A system of linear equations asks for values that satisfy all its equations at once.
Substitution or elimination can reveal a unique solution, no solution or infinitely many solutions.
Exercises with solutions
A linear system consists of two equations with two unknowns. With substitution you solve one equation for a variable and substitute it into the other. With elimination you rearrange an equation so that adding removes one variable.
Exercise 1
Solve: x + y = 10 and x − y = 4.
Show solution and working
x = 7, y = 3
Add both equations: 2x = 14, so x = 7. Substituting gives 7 + y = 10, so y = 3.
Exercise 2
Solve: y = 2x and x + y = 9.
Show solution and working
x = 3, y = 6
Substitute y = 2x: x + 2x = 9, so 3x = 9 and x = 3. Then y = 2 · 3 = 6.
Exercise 3
Solve: 2x + y = 11 and y = x + 2.
Show solution and working
x = 3, y = 5
Substituting: 2x + x + 2 = 11, so 3x = 9 and x = 3. Then y = 3 + 2 = 5.
Exercise 4
How many solutions can a linear system with two unknowns have?
Show solution and working
exactly one, none, or infinitely many
The two corresponding lines can meet in one point, be parallel, or be identical.
Common mistake
Working out only one variable and forgetting the second. A solution pair always consists of both values, and the check must hold in both equations.
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