Solving Systems of Equations

Substitution

Solve the following systems of equations using substitution.

1

$x+y=2$ $x-y=0$

2

$x^2+y^2=169$ $3x+2y=39$

3

$x=y+3$ $x=y^2+7$

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Elimination

Solve the following systems of equations using elimination.

1

$x+3y=-5$ $-x-8y=0$

2

$3x-2y=0$ $3x+2(y+5)=10$

3

$7x+12y=63$ $2x+3y=15$

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Back Substitution

Use back-substitution to solve the following systems of linear equations

1

$x-4y+3z=3$ $-y+z=-1$ $z=-5$

2

$x-2y+z=-6$ $2x-3y=-7$ $-x+3y-3z=11$

3

$x-2y+z=5$ $2x+3y+z=5$ $x+y+2z=3$

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Partial Fraction Decomposition

Write the partial fraction decomposition for the rational expression.

1

$\frac{4-x}{x^2+6x+8}$

2

$\frac{x^2+2x}{x^3-x^2+x-1}$

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Matrices

Determine the dimension of the following matrices:

1

$\begin{bmatrix} -3 \\ 1 \\ 10 \end{bmatrix}$

2

$\begin{bmatrix} 4 \end{bmatrix}$

3

$\begin{bmatrix} 5 & -1 & 0 & 0 \\ 3 & 2 & 2 & 9 \end{bmatrix}$

4

$\begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix}$

5

$\begin{bmatrix} 1 & 1 & 1 & 1 & 1 \end{bmatrix}$

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Gauss-Jordan Elimination

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination.

1

$x+y+4z=0$ $3x-y-z=0$ $-x+y-2z=-1$

2

$x-3y+z=2$ $3x-y-z=-6$ $-x-5y+3z=10$

3

$x+2y-z=3$ $x-y-z=-3$ $2x+y+3z=10$