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Systems of equationsAlgebraHigh schoolabout 1:11

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

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

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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. The problem
Solving an equation
$\begin{cases} x+ y = 3 \\ 2x+2y = 6 \end{cases}$
2. Steps, part 1
1. $\begin{cases} x + y = 3 \\ 2x + 2y = 6 \end{cases}$
  Write every equation in standard form
2. $\begin{bmatrix}1 & 1 & 3 \\ 2 & 2 & 6\end{bmatrix}$
  Build the augmented matrix [A | b]
3. Gauss–Jordan elimination
4. $R_{2} \to R_{2} - 2R_{1}:\ \begin{bmatrix}1 & 1 & 3 \\ 0 & 0 & 0\end{bmatrix}$
  Clear column 1 using the pivot in R1
3. Steps, part 2
1. $\begin{bmatrix}1 & 1 & 3 \\ 0 & 0 & 0\end{bmatrix}$
  Reduced row-echelon form
2. $x = 3 - t,\ y = t$
  y is a free variable
4. The answer
$x = 3 - t,\ y = t$
x + y = 3; 2x + 2y = 6 ⇒ Infinitely many solutions: x = 3 - t, y = t (t any real number)
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: x plus y equals 3; 2x plus 2y equals 6. 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. Clear column 1 using the pivot in R1.

Steps, part 2. 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 3 minus 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.

Solve the system x + y = 3, 2x + 2y = 6: Step-by-Step Lesson