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Systems of equationsAlgebraHigh schoolabout 1:08

Solve the system x - 2y = 1, 2x - 4y = 5: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: No solution (the system is inconsistent).

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    {x−2y=12x−4y=5\begin{cases} x-2y = 1 \\ 2x-4y = 5 \end{cases}
  2. 2. Steps, part 1
    1. {x−2y=12x−4y=5\begin{cases} x - 2y = 1 \\ 2x - 4y = 5 \end{cases}
      Write every equation in standard form
    2. [1−212−45]\begin{bmatrix}1 & -2 & 1 \\ 2 & -4 & 5\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R2→R2−2R1: [1−21003]R_{2} \to R_{2} - 2R_{1}:\ \begin{bmatrix}1 & -2 & 1 \\ 0 & 0 & 3\end{bmatrix}
      Clear column 1 using the pivot in R1
  3. 3. Steps, part 2
    1. R2→13R2: [1−21001]R_{2} \to \tfrac{1}{3} R_{2}:\ \begin{bmatrix}1 & -2 & 1 \\ 0 & 0 & 1\end{bmatrix}
      Divide R2 by 3
    2. R1→R1−1R2: [1−20001]R_{1} \to R_{1} - 1R_{2}:\ \begin{bmatrix}1 & -2 & 0 \\ 0 & 0 & 1\end{bmatrix}
      Clear column 3 using the pivot in R2
    3. [1−20001]\begin{bmatrix}1 & -2 & 0 \\ 0 & 0 & 1\end{bmatrix}
      Reduced row-echelon form
    4. A row reads 0 = non-zero
  4. 4. The answer
    No solution\text{No solution}

    x - 2y = 1; 2x - 4y = 5 ⇒ No solution (the system is inconsistent)

  5. 5. Check
    • The coefficient rows are dependent but the constants are not, which is the signature of parallel lines or planes.

Transcript

The problem. Here is the problem: x minus 2y equals 1; 2x minus 4y equals 5. 2 linear equations in 2 unknowns (x, y).

Steps, part 1. Write every equation in standard form. Move all variable terms to the left and constants to the right. Build the augmented matrix [A | b]. Columns: x, y | constant. Gauss–Jordan elimination. Use row operations to make each pivot 1 and clear the rest of its column. Clear column 1 using the pivot in R1.

Steps, part 2. Divide R2 by 3. Clear column 3 using the pivot in R2. Reduced row-echelon form. A row reads 0 equals non-zero. Row 2 says 0 equals 1, which is impossible, so the equations contradict each other.

The answer. So the answer is No solution (the system is inconsistent). The equations contradict each other, so no values satisfy them all at once.

Check. The coefficient rows are dependent but the constants are not, which is the signature of parallel lines or planes.

About systems of equations

Solve systems of two or three linear equations by elimination and substitution.

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Solve the system x - 2y = 1, 2x - 4y = 5: Animated Lesson