Equations and inequalities

Systems of equations

A system of equations asks for values that make all equations true at once. Three methods solve every system in an entrance test; pick the one with the least arithmetic.

The rules

• Substitution: solve one equation for a variable and insert it into the other.
• Equating: solve both equations for the same variable and set them equal.
• Elimination: add multiples of the equations so one variable cancels.
• Check your solution in every original equation.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. 2x + y = 12 2x + 3y = 24 Find y.

   • a3
   • b7
   • c5
   • d6
   Show solution

   Answer: 6

   Scale one equation so a coefficient matches, subtract to eliminate, back-substitute.

   Eliminate one variable, solve for the other, substitute back.

2. 2. x + 2y = 17 2x + y = 19 Find y.

   • a5
   • b7
   • c6
   • d4
   Show solution

   Answer: 5

   Scale one equation so a coefficient matches, subtract to eliminate, back-substitute.

   Eliminate one variable, solve for the other, substitute back.

3. 3. 3x + 2y = 27 2x + 3y = 23 Find x.

   • a8
   • b6
   • c3
   • d7
   Show solution

   Answer: 7

   Scale one equation so a coefficient matches, subtract to eliminate, back-substitute.

   Eliminate one variable, solve for the other, substitute back.

4. 4. x + 2y − 2z = 7 x + y + 3z = 6 3x + 3y + z = 18 Find z.

   • a5
   • b0
   • c-1
   • d1
   Show solution

   Answer: 0

   Eliminate one variable from two pairs of equations, solve the 2×2 system that is left, substitute back.

   Solution: x = 5, y = 1, z = 0; check it in all three equations.

5. 5. 3x + 2y = 23 x + 2y = 13 Find x.

   • a4
   • b9
   • c5
   • d6
   Show solution

   Answer: 5

   Scale one equation so a coefficient matches, subtract to eliminate, back-substitute.

   Eliminate one variable, solve for the other, substitute back.

6. 6. 3x + 3y = 15 x + 2y = 8 Find y.

   • a2
   • b5
   • c3
   • d4
   Show solution

   Answer: 3

   Scale one equation so a coefficient matches, subtract to eliminate, back-substitute.

   Eliminate one variable, solve for the other, substitute back.

Common mistakes

• Substituting back into the equation you already used, which always looks right.
• Multiplying only one side of an equation in the elimination method.
• Stopping after one variable when the question asks for the other.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs