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Quizzes > Mathematics > Algebra

Systems of Equations Quiz: Test Your Skills!

Take this systems of equations quiz challenge and boost your problem-solving skills.

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration of math papers with equations and shapes on teal background promoting systems of equations quiz

Ready to conquer math with confidence? Dive into our quiz on systems of equations that challenges you to solve pairs of equations using substitution and elimination. Whether you're a high school student brushing up on algebra or a lifelong learner seeking a brain workout, this free systems of equations quiz is designed for you. You'll enjoy practice with systems of equations and discover tips for accuracy under pressure. Test your skills with our interactive systems of equations quiz and then level up with a quick linear equations in two variables quiz . Plus, explore extra scenarios for practice systems of equations to keep the momentum going. Boost your math confidence and ace every problem - start now!

Solve the system of equations: x + y = 5 and x - y = 1.
(3, 2)
(2, 3)
(1, 4)
(4, 1)
To solve the system, add the equations: (x + y) + (x - y) = 5 + 1, which gives 2x = 6, so x = 3. Substitute x = 3 into x + y = 5 to find y = 2. This unique solution satisfies both equations. Learn more here.
Solve the system of equations: 2x - y = 1 and x + y = 5.
(2, 3)
(3, 2)
(1, 2)
(4, 1)
Add the two equations to eliminate y: (2x - y) + (x + y) = 1 + 5 gives 3x = 6, so x = 2. Then substitute x back into x + y = 5 to find y = 3. This straightforward elimination method yields the solution (2, 3). See detailed steps.
Classify the system: x + y = 2 and 2x + 2y = 4.
Infinitely many solutions
No solution
One unique solution
Exactly two solutions
The second equation is just 2 times the first, so they represent the same line. Every point on x + y = 2 also satisfies 2x + 2y = 4. Therefore, there are infinitely many solutions. More on dependent systems.
Classify the system: x + y = 1 and x + y = 3.
No solution
Infinitely many solutions
One unique solution
Exactly two solutions
Both equations have the same left side but different right sides, so they represent parallel lines. Parallel lines never intersect, meaning there is no common solution. Hence, the system is inconsistent. Learn more about inconsistent systems.
Solve the system: 3x + y = 9 and x + 2y = 8.
(2, 3)
(3, 2)
(1, 4)
(4, 1)
From the first equation, y = 9 - 3x. Substitute into the second: x + 2(9 - 3x) = 8 gives x + 18 - 6x = 8, so -5x = -10 and x = 2. Then y = 9 - 3(2) = 3. Substitution method details.
One number is 4 more than another, and their sum is 20. What are the two numbers?
(12, 8)
(16, 4)
(8, 12)
(10, 10)
Let the smaller number be y and the other x = y + 4. Their sum y + (y + 4) = 20 gives 2y + 4 = 20, so y = 8 and x = 12. The correct pair is (12, 8). See more worked examples.
Use elimination to solve: 4x + 5y = 1 and 2x - 3y = 7.
(19/11, -13/11)
(13/11, 19/11)
(-19/11, 13/11)
(19/7, -13/7)
Multiply the second equation by 2 to get 4x - 6y = 14. Subtract the first equation: (4x - 6y) - (4x + 5y) = 14 - 1 gives -11y = 13, so y = -13/11. Substitute back to find x = 19/11. Elimination method explained.
Classify the system: 5x - 2y = 1 and -10x + 4y = 2.
No solution
Infinitely many solutions
One unique solution
Exactly two solutions
The second equation is -2 times the first left side, but the right sides differ (2 ? -2). This inconsistency means the lines are parallel and do not intersect, so there is no solution. Inconsistent systems guide.
Solve the system: x² + y = 11 and x + y = 5.
{(3, 2), (-2, 7)}
{(4, 1), (1, 4)}
{(2, 3), (3, 2)}
{(-3, 8), (8, -3)}
Substitute y = 5 - x into x² + y = 11 to get x² + 5 - x = 11, which simplifies to x² - x - 6 = 0. Factoring gives (x - 3)(x + 2) = 0, so x = 3 or x = -2; then y = 2 or y = 7. Quadratic substitution.
Solve the system using matrices: x + 2y - z = 1, 3x - y + 2z = 4, 2x + y + z = 2.
(3/2, -1/2, -1/2)
(1, -1, 2)
(2, 0, -1)
(1/2, 1/2, 1/2)
Using Gaussian elimination or matrix inversion, one finds x = 3/2, y = -1/2, z = -1/2. The row operations lead to a unique solution for the three variables. Matrix inverse method.
For which value of k does the system x + k y = 2 and 3x + 6y = 6 have infinitely many solutions?
k = 2
k = 6
k = 1
k = 3
Write the first equation multiplied by 3: 3x + 3k y = 6. To match 3x + 6y = 6, we need 3k = 6, so k = 2. In that case both equations define the same line. Dependent systems.
A store sells two types of pens for $3 and $2. If 5 pens cost $12, how many of each type were sold?
(4 pens at $3, 1 pen at $2)
(3 pens at $3, 2 pens at $2)
(2 pens at $3, 3 pens at $2)
(1 pen at $3, 4 pens at $2)
Let x = number of $3 pens and y = number of $2 pens. Then x + y = 5 and 3x + 2y = 12. Solving gives x = 4 and y = 1. Mixture and cost problems.
For positive x and y satisfying x² + y² = 10 and xy = 3, what is the value of x + y?
4
-4
2
-2
Compute (x + y)² = x² + 2xy + y² = 10 + 2·3 = 16, so x + y = ±4. Since x and y are positive, x + y = 4. Sum and product of roots.
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Study Outcomes

  1. Apply the Substitution Method -

    Practice solving two-variable equation pairs by isolating one variable and substituting its expression to find accurate solutions.

  2. Apply the Elimination Method -

    Master combining equations through addition or subtraction to eliminate a variable and solve systems of equations with greater efficiency.

  3. Select the Optimal Strategy -

    Analyze different equation setups to decide whether substitution or elimination is the most efficient approach for each systems of equations quiz question.

  4. Verify and Interpret Solutions -

    Check your answers for consistency, recognize special cases like no solution or infinite solutions, and interpret the results correctly.

  5. Identify Strengths and Weaknesses -

    Use instant feedback from the free systems of equations quiz to pinpoint areas where you excel and topics that need further practice.

  6. Enhance Algebraic Confidence -

    Build confidence in algebra by tackling varied questions in the quiz on systems of equations and monitoring your improvement over time.

Cheat Sheet

  1. Definition and Solution Types -

    A system of equations consists of two or more linear equations with the same variables, and its solution is the point(s) where the lines intersect. According to Khan Academy, there are three types of solutions: one unique solution (intersecting lines), infinitely many (coincident lines), or none (parallel lines). Understanding these categories helps you predict whether to expect a single pair (x, y), infinite pairs, or no intersection.

  2. Substitution Method -

    In substitution, you solve one equation for a variable (e.g., x = 2y + 3) and plug that expression into the other equation. This method, endorsed by MIT OpenCourseWare, is especially handy when one equation is already solved for a variable. A quick tip: choose the simpler equation first to minimize algebraic errors.

  3. Elimination (Addition) Method -

    Elimination involves multiplying equations to align coefficients and then adding or subtracting to cancel one variable - Purplemath suggests lining up x- and y-terms in columns for clarity. For example, multiply 2x + 3y = 6 by 2 to match 4x - y = 5, then subtract. This systematic approach reduces it to a single-variable equation, making the solution straightforward.

  4. Graphical Interpretation -

    Plotting each equation on the coordinate plane visualizes the system: the solution is where the two lines cross. As noted by the National Library of Education, sketching can reveal whether lines are parallel (no solution) or overlapping (infinite solutions). Remember the mnemonic "Cross to Find the Treasure" - the intersection is your "X marks the spot."

  5. Verification and Interpretation -

    Always substitute your solution back into both original equations to verify accuracy - this simple check ensures you didn't make algebraic slips. The National Council of Teachers of Mathematics (NCTM) recommends this confidence-boosting step. Finally, interpret results in context: a unique point, endless solutions, or none, and be ready to explain why each occurs.

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