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Systems of equationsAlgebraHigh schoolabout 1:13

Solve the system 4x - 5y = 34, 7x + 9y = -47: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: x = 1, y = -6.

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
    {4x−5y=347x+9y=−47\begin{cases} 4x-5y = 34 \\ 7x+9y = -47 \end{cases}
  2. 2. Steps, part 1
    1. {4x−5y=347x+9y=−47\begin{cases} 4x - 5y = 34 \\ 7x + 9y = -47 \end{cases}
      Write every equation in standard form
    2. [4−53479−47]\begin{bmatrix}4 & -5 & 34 \\ 7 & 9 & -47\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R1→14R1: [1−5417279−47]R_{1} \to \tfrac{1}{4} R_{1}:\ \begin{bmatrix}1 & -\frac{5}{4} & \frac{17}{2} \\ 7 & 9 & -47\end{bmatrix}
      Divide R1 by 4
  3. 3. Steps, part 2
    1. R2→R2−7R1: [1−541720714−2132]R_{2} \to R_{2} - 7R_{1}:\ \begin{bmatrix}1 & -\frac{5}{4} & \frac{17}{2} \\ 0 & \frac{71}{4} & -\frac{213}{2}\end{bmatrix}
      Clear column 1 using the pivot in R1
    2. R2→1714R2: [1−5417201−6]R_{2} \to \tfrac{1}{\frac{71}{4}} R_{2}:\ \begin{bmatrix}1 & -\frac{5}{4} & \frac{17}{2} \\ 0 & 1 & -6\end{bmatrix}
      Divide R2 by 71/4
    3. R1→R1+54R2: [10101−6]R_{1} \to R_{1} + \frac{5}{4} R_{2}:\ \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & -6\end{bmatrix}
      Clear column 2 using the pivot in R2
    4. [10101−6]\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & -6\end{bmatrix}
      Reduced row-echelon form
  4. 4. Steps, part 3
    1. x=1, y=−6x = 1,\ y = -6
      Read off the unique solution
  5. 5. The answer
    x=1, y=−6x = 1,\ y = -6

    4x - 5y = 34; 7x + 9y = -47 ⇒ x = 1, y = -6

  6. 6. Check
    • Equation 1: 4x - 5y = 34 ✓
    • Equation 2: 7x + 9y = -47 ✓

Transcript

The problem. Here is the problem: 4x minus 5y equals 34; 7x plus 9y equals negative 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 4.

Steps, part 2. Clear column 1 using the pivot in R1. Divide R2 by 71 over 4. 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 1, y equals negative 6. The system has exactly one solution; substituting it back satisfies every equation.

Check. Equation 1: 4x minus 5y equals 34, which checks out. Equation 2: 7x plus 9y equals negative 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 4x - 5y = 34, 7x + 9y = -47 (Lesson)