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Systems of equationsAlgebraHigh schoolabout 1:15

Solve the system 9x + 4y = -55 and -8x + 8y = -32: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: x = -3, y = -7.

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
    {9x+4y=−55−8x+8y=−32\begin{cases} 9x+4y = -55 \\ -8x+8y = -32 \end{cases}
  2. 2. Steps, part 1
    1. {9x+4y=−55−8x+8y=−32\begin{cases} 9x + 4y = -55 \\ -8x + 8y = -32 \end{cases}
      Write every equation in standard form
    2. [94−55−88−32]\begin{bmatrix}9 & 4 & -55 \\ -8 & 8 & -32\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R1→19R1: [149−559−88−32]R_{1} \to \tfrac{1}{9} R_{1}:\ \begin{bmatrix}1 & \frac{4}{9} & -\frac{55}{9} \\ -8 & 8 & -32\end{bmatrix}
      Divide R1 by 9
  3. 3. Steps, part 2
    1. R2→R2+8R1: [149−55901049−7289]R_{2} \to R_{2} + 8R_{1}:\ \begin{bmatrix}1 & \frac{4}{9} & -\frac{55}{9} \\ 0 & \frac{104}{9} & -\frac{728}{9}\end{bmatrix}
      Clear column 1 using the pivot in R1
    2. R2→11049R2: [149−55901−7]R_{2} \to \tfrac{1}{\frac{104}{9}} R_{2}:\ \begin{bmatrix}1 & \frac{4}{9} & -\frac{55}{9} \\ 0 & 1 & -7\end{bmatrix}
      Divide R2 by 104/9
    3. R1→R1−49R2: [10−301−7]R_{1} \to R_{1} - \frac{4}{9} R_{2}:\ \begin{bmatrix}1 & 0 & -3 \\ 0 & 1 & -7\end{bmatrix}
      Clear column 2 using the pivot in R2
    4. [10−301−7]\begin{bmatrix}1 & 0 & -3 \\ 0 & 1 & -7\end{bmatrix}
      Reduced row-echelon form
  4. 4. Steps, part 3
    1. x=−3, y=−7x = -3,\ y = -7
      Read off the unique solution
  5. 5. The answer
    x=−3, y=−7x = -3,\ y = -7

    9x + 4y = -55; -8x + 8y = -32 ⇒ x = -3, y = -7

  6. 6. Check
    • Equation 1: 9x + 4y = -55 ✓
    • Equation 2: -8x + 8y = -32 ✓

Transcript

The problem. Here is the problem: 9x plus 4y equals negative 55; negative 8x plus 8y equals negative 32. 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 9.

Steps, part 2. Clear column 1 using the pivot in R1. Divide R2 by 104 over 9. 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 3, y equals negative 7. The system has exactly one solution; substituting it back satisfies every equation.

Check. Equation 1: 9x plus 4y equals negative 55, which checks out. Equation 2: negative 8x plus 8y equals negative 32, 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 9x + 4y = -55 and -8x + 8y = -32 (Lesson)