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Systems of equationsAlgebraHigh schoolabout 1:13

Solve the system 9x + 5y = 15, -5x - 9y = 29: animated lesson

A narrated, step-by-step systems of equations video lesson with captions. Final answer: x = 5, y = -6.

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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
$\begin{cases} 9x+5y = 15 \\ -5x-9y = 29 \end{cases}$
2. Steps, part 1
1. $\begin{cases} 9x + 5y = 15 \\ -5x - 9y = 29 \end{cases}$
  Write every equation in standard form
2. $\begin{bmatrix}9 & 5 & 15 \\ -5 & -9 & 29\end{bmatrix}$
  Build the augmented matrix [A | b]
3. Gauss–Jordan elimination
4. $R_{1} \to \tfrac{1}{9} R_{1}:\ \begin{bmatrix}1 & \frac{5}{9} & \frac{5}{3} \\ -5 & -9 & 29\end{bmatrix}$
  Divide R1 by 9
3. Steps, part 2
1. $R_{2} \to R_{2} + 5R_{1}:\ \begin{bmatrix}1 & \frac{5}{9} & \frac{5}{3} \\ 0 & -\frac{56}{9} & \frac{112}{3}\end{bmatrix}$
  Clear column 1 using the pivot in R1
2. $R_{2} \to \tfrac{1}{-\frac{56}{9}} R_{2}:\ \begin{bmatrix}1 & \frac{5}{9} & \frac{5}{3} \\ 0 & 1 & -6\end{bmatrix}$
  Divide R2 by -56/9
3. $R_{1} \to R_{1} - \frac{5}{9} R_{2}:\ \begin{bmatrix}1 & 0 & 5 \\ 0 & 1 & -6\end{bmatrix}$
  Clear column 2 using the pivot in R2
4. $\begin{bmatrix}1 & 0 & 5 \\ 0 & 1 & -6\end{bmatrix}$
  Reduced row-echelon form
4. Steps, part 3
1. $x = 5,\ y = -6$
  Read off the unique solution
5. The answer
$x = 5,\ y = -6$
9x + 5y = 15; -5x - 9y = 29 ⇒ x = 5, y = -6
6. Check
- Equation 1: 9x + 5y = 15 ✓
- Equation 2: -5x - 9y = 29 ✓

Transcript

The problem. Here is the problem: 9x plus 5y equals 15; negative 5x minus 9y equals 29. 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 9.

Steps, part 2. Clear column 1 using the pivot in R1. Divide R2 by negative 56 over 9. 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 negative 6. The system has exactly one solution; substituting it back satisfies every equation.

Check. Equation 1: 9x plus 5y equals 15, which checks out. Equation 2: negative 5x minus 9y equals 29, which checks out.

About systems of equations

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

Solve the system 9x + 5y = 15, -5x - 9y = 29 (Lesson)