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Systems of equationsAlgebraHigh schoolabout 1:12

Solve the system -3x + 9y = 21, -2x - 4y = -6: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: x = -1, y = 2.

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
    {−3x+9y=21−2x−4y=−6\begin{cases} -3x+9y = 21 \\ -2x-4y = -6 \end{cases}
  2. 2. Steps, part 1
    1. {−3x+9y=21−2x−4y=−6\begin{cases} -3x + 9y = 21 \\ -2x - 4y = -6 \end{cases}
      Write every equation in standard form
    2. [−3921−2−4−6]\begin{bmatrix}-3 & 9 & 21 \\ -2 & -4 & -6\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R1→1−3R1: [1−3−7−2−4−6]R_{1} \to \tfrac{1}{-3} R_{1}:\ \begin{bmatrix}1 & -3 & -7 \\ -2 & -4 & -6\end{bmatrix}
      Divide R1 by -3
  3. 3. Steps, part 2
    1. R2→R2+2R1: [1−3−70−10−20]R_{2} \to R_{2} + 2R_{1}:\ \begin{bmatrix}1 & -3 & -7 \\ 0 & -10 & -20\end{bmatrix}
      Clear column 1 using the pivot in R1
    2. R2→1−10R2: [1−3−7012]R_{2} \to \tfrac{1}{-10} R_{2}:\ \begin{bmatrix}1 & -3 & -7 \\ 0 & 1 & 2\end{bmatrix}
      Divide R2 by -10
    3. R1→R1+3R2: [10−1012]R_{1} \to R_{1} + 3R_{2}:\ \begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 2\end{bmatrix}
      Clear column 2 using the pivot in R2
    4. [10−1012]\begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 2\end{bmatrix}
      Reduced row-echelon form
  4. 4. Steps, part 3
    1. x=−1, y=2x = -1,\ y = 2
      Read off the unique solution
  5. 5. The answer
    x=−1, y=2x = -1,\ y = 2

    -3x + 9y = 21; -2x - 4y = -6 ⇒ x = -1, y = 2

  6. 6. Check
    • Equation 1: -3x + 9y = 21 ✓
    • Equation 2: -2x - 4y = -6 ✓

Transcript

The problem. Here is the problem: negative 3x plus 9y equals 21; negative 2x minus 4y equals negative 6. 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. Divide R1 by negative 3.

Steps, part 2. Clear column 1 using the pivot in R1. Divide R2 by negative 10. Clear column 2 using the pivot in R2. Reduced row-echelon form.

Steps, part 3. Read off the unique solution.

The answer. So the answer is x equals negative 1, y equals 2. The system has exactly one solution; substituting it back satisfies every equation.

Check. Equation 1: negative 3x plus 9y equals 21, which checks out. Equation 2: negative 2x minus 4y equals negative 6, which checks out.

About systems of equations

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

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Solve the system -3x + 9y = 21, -2x - 4y = -6 (Lesson)