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Systems of Equations and Equivalent Systems

A system is a group of equations that must all be true together. Two systems are equivalent when they have exactly the same solutions. Swapping equations, multiplying one by a number that is not zero, and adding a multiple of one equation to another all keep a system equivalent, so we can make it simpler without losing the answer.

๐ŸŽฌ Step-by-step story

  1. Look at one equation, x + y = 5. Every point on this red line makes it true. There are endless answers.
  2. Add a second equation, x โˆ’ y = 1, the blue line. A system means both must be true at once.
  3. Only one point sits on both lines: (3, 2). That point is the solution of the system.
  4. Multiply the first equation by 2. You get 2x + 2y = 10. The line does not move, so the answer is the same.
  5. Now add the two equations. You get 2x = 6, which is the line x = 3. It still passes through (3, 2), so the new system is equivalent and much easier.
  6. Your turn. Slide k to change equation 2 into (equation 2) + k ร— (equation 1). Watch the line turn but always stay on the point. Then try the multiply by 0 button.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

๐Ÿค” Common doubts, cleared

Can one equation have just one solution?

No. One equation with two unknowns has endless solutions, a whole line of points. A second equation picks one point.

What if the lines never cross?

Then the system has no solution. The lines are parallel.

Why is the crossing point the solution?

It is the only point on both lines, so it makes both equations true.

Does multiplying change the answer?

No, as long as you multiply the whole equation by a number that is not 0. The line is the same line.

Why add the equations at all?

It cancels one unknown, so you get a one-unknown equation that is easy to solve.

Why can the multiply by 0 move be bad?

It turns the equation into 0 = 0, which is true for every point. You lose the line and the system gets endless solutions.

What is a system of equations?

A system is a set of equations with the same unknowns. A solution must make every equation true at the same time. For two lines, the solution is the point where the lines cross. Two lines can cross at one point (one solution), never meet because they are parallel (no solution), or be the same line (endless solutions).

Example: x + y = 5 and x โˆ’ y = 1. Try (3, 2): 3 + 2 = 5 and 3 โˆ’ 2 = 1. Both are true, so (3, 2) is the solution.

What does equivalent mean?

Two systems are equivalent if they have exactly the same set of solutions. Not one extra, not one missing. They can look very different. For example, {x + y = 5, x โˆ’ y = 1} and {x = 3, y = 2} are equivalent, because both have only the solution (3, 2).

We change a hard system into an equivalent easy one, so the answer we find is also right for the hard one.

Safe moves (equivalent transformations)

Unsafe moves: multiply by 0 (the equation becomes 0 = 0 and says nothing), or change only one side of an equation. Squaring both sides can also add false answers, so you must check at the end.

Solving by elimination

Elimination is a chain of safe moves that removes one unknown.

  1. Write the system. Example: 2x + 3y = 12 and 4x โˆ’ y = 10.
  2. Multiply the second equation by 3: 12x โˆ’ 3y = 30.
  3. Add it to the first: 14x = 42, so x = 3.
  4. Put x = 3 in 4x โˆ’ y = 10: 12 โˆ’ y = 10, so y = 2.
  5. Check in both original equations.

Each new system is equivalent to the one before, so (3, 2) is the answer to the first one too.

Try it

Predict, then check: in the 3D free play move the slider k. Before you move it, say what you think will happen to the crossing point. Then find the k that makes the blue line vertical. Next press the multiply by 0 button and see why it is not a safe move.

Key formulas and definitions

Worked examples

1. Solve x + y = 7 and x โˆ’ y = 1.

Add the equations: 2x = 8, so x = 4. Then y = 7 โˆ’ 4 = 3. Check: 4 + 3 = 7 and 4 โˆ’ 3 = 1. Solution (4, 3).

2. Solve 3x + y = 10 and x + y = 4.

Subtract the second from the first: 2x = 6, so x = 3. Then y = 4 โˆ’ 3 = 1. Check: 9 + 1 = 10 and 3 + 1 = 4. Solution (3, 1).

3. Show that {x + y = 5, x โˆ’ y = 1} is equivalent to {x = 3, x + y = 5}.

Add the two original equations to get 2x = 6, so x = 3. Keep x + y = 5. The new system came from safe moves, and its solution is x = 3, y = 2. Both systems have only (3, 2).

4. Solve 2x + 3y = 12 and 4x โˆ’ y = 10.

Multiply the second equation by 3: 12x โˆ’ 3y = 30. Add to the first: 14x = 42, so x = 3. Then 4(3) โˆ’ y = 10 gives y = 2. Solution (3, 2).

5. In the system x + y = 5, x โˆ’ y = 1, replace equation 2 by (equation 2) + 2 ร— (equation 1). Write the new equation and check the old solution.

New equation: (x โˆ’ y) + 2(x + y) = 1 + 10, so 3x + y = 11. Check (3, 2): 9 + 2 = 11. True, so the solution survives.

Common mistakes

Practice quiz

1. Two systems are equivalent when they have:
2. Which move is NOT safe?
3. Adding x + y = 5 and x โˆ’ y = 1 gives:
4. Two parallel lines as a system have:
5. The solution of {x + y = 7, x โˆ’ y = 1} is:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is an equivalent system in simple words?

It is a new system that has the very same answers as the old one, only written in an easier way.

Does elimination always give an equivalent system?

Yes, as long as you only swap, multiply by a number that is not zero, or add a multiple of one equation to another.

Why do we check the answer at the end?

Checking catches arithmetic slips and, when you used squaring or other risky steps, removes extra false answers.

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