Algebra Examples

$[\begin{array}{c} 3 & - 2 \\ - 5 & 4 \end{array}] x - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 1

Reorder factors in $[\begin{array}{c} 3 & - 2 \\ - 5 & 4 \end{array}] x - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$ .

$x [\begin{array}{c} 3 & - 2 \\ - 5 & 4 \end{array}] - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x$ by each element of the matrix.

$[\begin{array}{c} x \cdot 3 & x \cdot - 2 \\ x \cdot - 5 & x \cdot 4 \end{array}] - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 2.1.2

Simplify each element in the matrix.

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Step 2.1.2.1

Move $3$ to the left of $x$ .

$[\begin{array}{c} 3 x & x \cdot - 2 \\ x \cdot - 5 & x \cdot 4 \end{array}] - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 2.1.2.2

Move $- 2$ to the left of $x$ .

$[\begin{array}{c} 3 x & - 2 x \\ x \cdot - 5 & x \cdot 4 \end{array}] - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 2.1.2.3

Move $- 5$ to the left of $x$ .

$[\begin{array}{c} 3 x & - 2 x \\ - 5 x & x \cdot 4 \end{array}] - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 2.1.2.4

Move $4$ to the left of $x$ .

$[\begin{array}{c} 3 x & - 2 x \\ - 5 x & 4 x \end{array}] - [\begin{array}{c} 3 & - 1 \\ 6 & 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 2.2

Subtract the corresponding elements.

$[\begin{array}{c} 3 x - 3 & - 2 x + 1 \\ - 5 x - 6 & 4 x - 2 \end{array}] = [\begin{array}{c} 0 & - 4 \\ - 2 & 3 \end{array}]$

Step 3

Write as a linear system of equations.

$3 x - 3 = 0$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4

Solve the system of equations.

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Step 4.1

Solve for $x$ in $3 x - 3 = 0$ .

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Step 4.1.1

Add $3$ to both sides of the equation.

$3 x = 3$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.1.2

Divide each term in $3 x = 3$ by $3$ and simplify.

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Step 4.1.2.1

Divide each term in $3 x = 3$ by $3$ .

$\frac{3 x}{3} = \frac{3}{3}$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{3}{3}$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{3}$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$x = \frac{3}{3}$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$x = \frac{3}{3}$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $3$ by $3$ .

$x = 1$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$x = 1$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$x = 1$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$x = 1$

$- 2 x + 1 = - 4$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.2

Replace all occurrences of $x$ with $1$ in each equation.

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Step 4.2.1

Replace all occurrences of $x$ in $- 2 x + 1 = - 4$ with $1$ .

$- 2 \cdot 1 + 1 = - 4$

$x = 1$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify $- 2 \cdot 1 + 1$ .

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Step 4.2.2.1.1

Multiply $- 2$ by $1$ .

$- 2 + 1 = - 4$

$x = 1$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.2.2.1.2

Add $- 2$ and $1$ .

$- 1 = - 4$

$x = 1$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$- 1 = - 4$

$x = 1$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

$- 1 = - 4$

$x = 1$

$- 5 x - 6 = - 2$

$4 x - 2 = 3$

Step 4.2.3

Replace all occurrences of $x$ in $- 5 x - 6 = - 2$ with $1$ .

$- 5 \cdot 1 - 6 = - 2$

$- 1 = - 4$

$x = 1$

$4 x - 2 = 3$

Step 4.2.4

Simplify the left side.

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Step 4.2.4.1

Simplify $- 5 \cdot 1 - 6$ .

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Step 4.2.4.1.1

Multiply $- 5$ by $1$ .

$- 5 - 6 = - 2$

$- 1 = - 4$

$x = 1$

$4 x - 2 = 3$

Step 4.2.4.1.2

Subtract $6$ from $- 5$ .

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

$4 x - 2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

$4 x - 2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

$4 x - 2 = 3$

Step 4.2.5

Replace all occurrences of $x$ in $4 x - 2 = 3$ with $1$ .

$4 (1) - 2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

Step 4.2.6

Simplify the left side.

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Step 4.2.6.1

Simplify $4 (1) - 2$ .

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Step 4.2.6.1.1

Multiply $4$ by $1$ .

$4 - 2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

Step 4.2.6.1.2

Subtract $2$ from $4$ .

$2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

$2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

$2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

$2 = 3$

$- 11 = - 2$

$- 1 = - 4$

$x = 1$

Step 4.3

Since $2 = 3$ is not true, there is no solution.

$No solution$