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Systems of equationsAlgebraHigh schoolabout 1:16

Solve the system 2x + 3y = 7, 4x + 6y = 14: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: Infinitely many solutions: x = 7/2 - (3/2)t, y = t (t any real number).

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
    {2x+3y=74x+6y=14\begin{cases} 2x+3y = 7 \\ 4x+6y = 14 \end{cases}
  2. 2. Steps, part 1
    1. {2x+3y=74x+6y=14\begin{cases} 2x + 3y = 7 \\ 4x + 6y = 14 \end{cases}
      Write every equation in standard form
    2. [2374614]\begin{bmatrix}2 & 3 & 7 \\ 4 & 6 & 14\end{bmatrix}
      Build the augmented matrix [A | b]
    3. Gauss–Jordan elimination
    4. R1→12R1: [132724614]R_{1} \to \tfrac{1}{2} R_{1}:\ \begin{bmatrix}1 & \frac{3}{2} & \frac{7}{2} \\ 4 & 6 & 14\end{bmatrix}
      Divide R1 by 2
  3. 3. Steps, part 2
    1. R2→R2−4R1: [13272000]R_{2} \to R_{2} - 4R_{1}:\ \begin{bmatrix}1 & \frac{3}{2} & \frac{7}{2} \\ 0 & 0 & 0\end{bmatrix}
      Clear column 1 using the pivot in R1
    2. [13272000]\begin{bmatrix}1 & \frac{3}{2} & \frac{7}{2} \\ 0 & 0 & 0\end{bmatrix}
      Reduced row-echelon form
    3. x=7/2−(3/2)t, y=tx = 7/2 - (3/2)t,\ y = t
      y is a free variable
  4. 4. The answer
    x=7/2−(3/2)t, y=tx = 7/2 - (3/2)t,\ y = t

    2x + 3y = 7; 4x + 6y = 14 ⇒ Infinitely many solutions: x = 7/2 - (3/2)t, y = t (t any real number)

  5. 5. Check
    • Rank 1 < 2 unknowns with no contradictory row, so the solution set is a 1-parameter family.

Transcript

The problem. Here is the problem: 2x plus 3y equals 7; 4x plus 6y equals 14. 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 2.

Steps, part 2. Clear column 1 using the pivot in R1. Reduced row-echelon form. y is a free variable. There are 1 pivot for 2 unknowns, so 1 variable is free. Let y equals t and solve the pivot rows for the rest.

The answer. So the answer is Infinitely many solutions: x equals 7 over 2 minus (3 over 2)t, y equals t (t any real number). The equations are dependent, so they describe the same line/plane (or an intersection that is a whole line), giving infinitely many solutions.

Check. Rank 1 less than 2 unknowns with no contradictory row, so the solution set is a 1-parameter family.

About systems of equations

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

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Solve the system 2x + 3y = 7, 4x + 6y = 14: Animated Lesson