Algebra Examples

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} 1 & 3 \\ 4 & 3 \end{array}] x = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 1

Reorder factors in $- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} 1 & 3 \\ 4 & 3 \end{array}] x = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$ .

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + x [\begin{array}{c} 1 & 3 \\ 4 & 3 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

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.

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} x \cdot 1 & x \cdot 3 \\ x \cdot 4 & x \cdot 3 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.1.2

Simplify each element in the matrix.

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Step 2.1.2.1

Multiply $x$ by $1$ .

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} x & x \cdot 3 \\ x \cdot 4 & x \cdot 3 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.1.2.2

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

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} x & 3 x \\ x \cdot 4 & x \cdot 3 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.1.2.3

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

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} x & 3 x \\ 4 x & x \cdot 3 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.1.2.4

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

$- [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] + [\begin{array}{c} x & 3 x \\ 4 x & 3 x \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.2

Reorder the matrices.

$[\begin{array}{c} x & 3 x \\ 4 x & 3 x \end{array}] - [\begin{array}{c} - 4 & 0 \\ 4 & - 1 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.3

Subtract the corresponding elements.

$[\begin{array}{c} x + 4 & 3 x - 0 \\ 4 x - 4 & 3 x + 1 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 2.4

Subtract $0$ from $3 x$ .

$[\begin{array}{c} x + 4 & 3 x \\ 4 x - 4 & 3 x + 1 \end{array}] = [\begin{array}{c} 13 & 12 \\ 23 & 13 \end{array}]$

Step 3

Write as a linear system of equations.

$x + 4 = 13$

$3 x = 12$

$4 x - 4 = 23$

$3 x + 1 = 13$

Step 4

Solve the system of equations.

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Step 4.1

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

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Step 4.1.1

Subtract $4$ from both sides of the equation.

$x = 13 - 4$

$3 x = 12$

$4 x - 4 = 23$

$3 x + 1 = 13$

Step 4.1.2

Subtract $4$ from $13$ .

$x = 9$

$3 x = 12$

$4 x - 4 = 23$

$3 x + 1 = 13$

$x = 9$

$3 x = 12$

$4 x - 4 = 23$

$3 x + 1 = 13$

Step 4.2

Replace all occurrences of $x$ with $9$ in each equation.

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Step 4.2.1

Replace all occurrences of $x$ in $3 x = 12$ with $9$ .

$3 (9) = 12$

$x = 9$

$4 x - 4 = 23$

$3 x + 1 = 13$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Multiply $3$ by $9$ .

$27 = 12$

$x = 9$

$4 x - 4 = 23$

$3 x + 1 = 13$

$27 = 12$

$x = 9$

$4 x - 4 = 23$

$3 x + 1 = 13$

Step 4.2.3

Replace all occurrences of $x$ in $4 x - 4 = 23$ with $9$ .

$4 (9) - 4 = 23$

$27 = 12$

$x = 9$

$3 x + 1 = 13$

Step 4.2.4

Simplify the left side.

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Step 4.2.4.1

Simplify $4 (9) - 4$ .

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Step 4.2.4.1.1

Multiply $4$ by $9$ .

$36 - 4 = 23$

$27 = 12$

$x = 9$

$3 x + 1 = 13$

Step 4.2.4.1.2

Subtract $4$ from $36$ .

$32 = 23$

$27 = 12$

$x = 9$

$3 x + 1 = 13$

$32 = 23$

$27 = 12$

$x = 9$

$3 x + 1 = 13$

$32 = 23$

$27 = 12$

$x = 9$

$3 x + 1 = 13$

Step 4.2.5

Replace all occurrences of $x$ in $3 x + 1 = 13$ with $9$ .

$3 (9) + 1 = 13$

$32 = 23$

$27 = 12$

$x = 9$

Step 4.2.6

Simplify the left side.

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Step 4.2.6.1

Simplify $3 (9) + 1$ .

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Step 4.2.6.1.1

Multiply $3$ by $9$ .

$27 + 1 = 13$

$32 = 23$

$27 = 12$

$x = 9$

Step 4.2.6.1.2

Add $27$ and $1$ .

$28 = 13$

$32 = 23$

$27 = 12$

$x = 9$

$28 = 13$

$32 = 23$

$27 = 12$

$x = 9$

$28 = 13$

$32 = 23$

$27 = 12$

$x = 9$

$28 = 13$

$32 = 23$

$27 = 12$

$x = 9$

Step 4.3

Since $28 = 13$ is not true, there is no solution.

$No solution$