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Systems of equationsAlgebraHigh schoolabout 1:09

Solve the system 5x - 7y = 28, x - y = 4: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: x = 0, y = -4.

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
    {5x−7y=28x−y=4\begin{cases} 5x-7y = 28 \\ x- y = 4 \end{cases}
  2. 2. Steps, part 1
    1. {5x−7y=28x−y=4\begin{cases} 5x - 7y = 28 \\ x - y = 4 \end{cases}
      Write every equation in standard form
    2. [5−7281−14]\begin{bmatrix}5 & -7 & 28 \\ 1 & -1 & 4\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R1→15R1: [1−752851−14]R_{1} \to \tfrac{1}{5} R_{1}:\ \begin{bmatrix}1 & -\frac{7}{5} & \frac{28}{5} \\ 1 & -1 & 4\end{bmatrix}
      Divide R1 by 5
  3. 3. Steps, part 2
    1. R2→R2−1R1: [1−75285025−85]R_{2} \to R_{2} - 1R_{1}:\ \begin{bmatrix}1 & -\frac{7}{5} & \frac{28}{5} \\ 0 & \frac{2}{5} & -\frac{8}{5}\end{bmatrix}
      Clear column 1 using the pivot in R1
    2. R2→125R2: [1−7528501−4]R_{2} \to \tfrac{1}{\frac{2}{5}} R_{2}:\ \begin{bmatrix}1 & -\frac{7}{5} & \frac{28}{5} \\ 0 & 1 & -4\end{bmatrix}
      Divide R2 by 2/5
    3. R1→R1+75R2: [10001−4]R_{1} \to R_{1} + \frac{7}{5} R_{2}:\ \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & -4\end{bmatrix}
      Clear column 2 using the pivot in R2
    4. [10001−4]\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & -4\end{bmatrix}
      Reduced row-echelon form
  4. 4. Steps, part 3
    1. x=0, y=−4x = 0,\ y = -4
      Read off the unique solution
  5. 5. The answer
    x=0, y=−4x = 0,\ y = -4

    5x - 7y = 28; x - y = 4 ⇒ x = 0, y = -4

  6. 6. Check
    • Equation 1: 5x - 7y = 28 ✓
    • Equation 2: x - y = 4 ✓

Transcript

The problem. Here is the problem: 5x minus 7y equals 28; x minus y equals 4. 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 5.

Steps, part 2. Clear column 1 using the pivot in R1. Divide R2 by 2 over 5. 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 0, y equals negative 4. The system has exactly one solution; substituting it back satisfies every equation.

Check. Equation 1: 5x minus 7y equals 28, which checks out. Equation 2: x minus y equals 4, 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 5x - 7y = 28, x - y = 4: Animated Lesson