Linear Algebra Examples

$[\begin{array}{c} c + d \\ 2 c - d \\ d - 2 e \end{array}] = [\begin{array}{c} 5 \\ 4 \\ 6 \end{array}]$

Step 1

Write as a linear system of equations.

$c + d = 5$

$2 c - d = 4$

$d - 2 e = 6$

Step 2

Solve the system of equations.

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Step 2.1

Add $2 e$ to both sides of the equation.

$d = 6 + 2 e$

$c + d = 5$

$2 c - d = 4$

Step 2.2

Replace all occurrences of $d$ with $6 + 2 e$ in each equation.

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Step 2.2.1

Replace all occurrences of $d$ in $c + d = 5$ with $6 + 2 e$ .

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c - d = 4$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Remove parentheses.

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c - d = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c - d = 4$

Step 2.2.3

Replace all occurrences of $d$ in $2 c - d = 4$ with $6 + 2 e$ .

$2 c - (6 + 2 e) = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.2.4

Simplify the left side.

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Apply the distributive property.

$2 c - 1 \cdot 6 - (2 e) = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.2.4.1.2

Multiply $- 1$ by $6$ .

$2 c - 6 - (2 e) = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.2.4.1.3

Multiply $2$ by $- 1$ .

$2 c - 6 - 2 e = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c - 6 - 2 e = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c - 6 - 2 e = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c - 6 - 2 e = 4$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3

Solve for $c$ in $2 c - 6 - 2 e = 4$ .

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Step 2.3.1

Move all terms not containing $c$ to the right side of the equation.

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Step 2.3.1.1

Add $6$ to both sides of the equation.

$2 c - 2 e = 4 + 6$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.1.2

Add $2 e$ to both sides of the equation.

$2 c = 4 + 6 + 2 e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.1.3

Add $4$ and $6$ .

$2 c = 10 + 2 e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$2 c = 10 + 2 e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.2

Divide each term in $2 c = 10 + 2 e$ by $2$ and simplify.

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Step 2.3.2.1

Divide each term in $2 c = 10 + 2 e$ by $2$ .

$\frac{2 c}{2} = \frac{10}{2} + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{2 c}{2} = \frac{10}{2} + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{10}{2} + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = \frac{10}{2} + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = \frac{10}{2} + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Simplify each term.

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Step 2.3.2.3.1.1

Divide $10$ by $2$ .

$c = 5 + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.2.3.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.3.1.2.1

Cancel the common factor.

$c = 5 + \frac{2 e}{2}$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.3.2.3.1.2.2

Divide $e$ by $1$ .

$c = 5 + e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = 5 + e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = 5 + e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = 5 + e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = 5 + e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

$c = 5 + e$

$c + 6 + 2 e = 5$

$d = 6 + 2 e$

Step 2.4

Replace all occurrences of $c$ with $5 + e$ in each equation.

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Step 2.4.1

Replace all occurrences of $c$ in $c + 6 + 2 e = 5$ with $5 + e$ .

$(5 + e) + 6 + 2 e = 5$

$c = 5 + e$

$d = 6 + 2 e$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify $(5 + e) + 6 + 2 e$ .

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Step 2.4.2.1.1

Add $5$ and $6$ .

$11 + e + 2 e = 5$

$c = 5 + e$

$d = 6 + 2 e$

Step 2.4.2.1.2

Add $e$ and $2 e$ .

$11 + 3 e = 5$

$c = 5 + e$

$d = 6 + 2 e$

$11 + 3 e = 5$

$c = 5 + e$

$d = 6 + 2 e$

$11 + 3 e = 5$

$c = 5 + e$

$d = 6 + 2 e$

$11 + 3 e = 5$

$c = 5 + e$

$d = 6 + 2 e$

Step 2.5

Since $11 + 3 e = 5$ is not true, there is no solution.

$No solution$