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Systems of equationsAlgebraHigh schoolabout 1:10

Solve the system 3x + 7y = 22, 8x + 7y = 47: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: x = 5, y = 1.

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+7y=228x+7y=47\begin{cases} 3x+7y = 22 \\ 8x+7y = 47 \end{cases}
  2. 2. Steps, part 1
    1. {3x+7y=228x+7y=47\begin{cases} 3x + 7y = 22 \\ 8x + 7y = 47 \end{cases}
      Write every equation in standard form
    2. [37228747]\begin{bmatrix}3 & 7 & 22 \\ 8 & 7 & 47\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R1→13R1: [1732238747]R_{1} \to \tfrac{1}{3} R_{1}:\ \begin{bmatrix}1 & \frac{7}{3} & \frac{22}{3} \\ 8 & 7 & 47\end{bmatrix}
      Divide R1 by 3
  3. 3. Steps, part 2
    1. R2→R2−8R1: [1732230−353−353]R_{2} \to R_{2} - 8R_{1}:\ \begin{bmatrix}1 & \frac{7}{3} & \frac{22}{3} \\ 0 & -\frac{35}{3} & -\frac{35}{3}\end{bmatrix}
      Clear column 1 using the pivot in R1
    2. R2→1−353R2: [173223011]R_{2} \to \tfrac{1}{-\frac{35}{3}} R_{2}:\ \begin{bmatrix}1 & \frac{7}{3} & \frac{22}{3} \\ 0 & 1 & 1\end{bmatrix}
      Divide R2 by -35/3
    3. R1→R1−73R2: [105011]R_{1} \to R_{1} - \frac{7}{3} R_{2}:\ \begin{bmatrix}1 & 0 & 5 \\ 0 & 1 & 1\end{bmatrix}
      Clear column 2 using the pivot in R2
    4. [105011]\begin{bmatrix}1 & 0 & 5 \\ 0 & 1 & 1\end{bmatrix}
      Reduced row-echelon form
  4. 4. Steps, part 3
    1. x=5, y=1x = 5,\ y = 1
      Read off the unique solution
  5. 5. The answer
    x=5, y=1x = 5,\ y = 1

    3x + 7y = 22; 8x + 7y = 47 ⇒ x = 5, y = 1

  6. 6. Check
    • Equation 1: 3x + 7y = 22 ✓
    • Equation 2: 8x + 7y = 47 ✓

Transcript

The problem. Here is the problem: 3x plus 7y equals 22; 8x plus 7y equals 47. 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 3.

Steps, part 2. Clear column 1 using the pivot in R1. Divide R2 by negative 35 over 3. 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 5, y equals 1. The system has exactly one solution; substituting it back satisfies every equation.

Check. Equation 1: 3x plus 7y equals 22, which checks out. Equation 2: 8x plus 7y equals 47, 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 + 7y = 22, 8x + 7y = 47: Animated Lesson