Algebra Examples

$[\begin{array}{c} a & 2 b \\ c - 2 & d + 3 \end{array}] = [\begin{array}{c} 5 & - 7 \\ 10 & 10 \end{array}]$

Step 1

Write as a linear system of equations.

$a = 5$

$2 b = - 7$

$c - 2 = 10$

$d + 3 = 10$

Step 2

Solve the system of equations.

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Step 2.1

Divide each term in $2 b = - 7$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 b = - 7$ by $2$ .

$\frac{2 b}{2} = \frac{- 7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{- 7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

Step 2.1.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

$b = \frac{- 7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

$b = \frac{- 7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Move the negative in front of the fraction.

$b = - \frac{7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

$b = - \frac{7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

$b = - \frac{7}{2}$

$a = 5$

$c - 2 = 10$

$d + 3 = 10$

Step 2.2

Replace all occurrences of $b$ with $- \frac{7}{2}$ in each equation.

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Step 2.2.1

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

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Step 2.2.1.1

Add $2$ to both sides of the equation.

$c = 10 + 2$

$b = - \frac{7}{2}$

$a = 5$

$d + 3 = 10$

Step 2.2.1.2

Add $10$ and $2$ .

$c = 12$

$b = - \frac{7}{2}$

$a = 5$

$d + 3 = 10$

$c = 12$

$b = - \frac{7}{2}$

$a = 5$

$d + 3 = 10$

Step 2.2.2

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

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Step 2.2.2.1

Subtract $3$ from both sides of the equation.

$d = 10 - 3$

$c = 12$

$b = - \frac{7}{2}$

$a = 5$

Step 2.2.2.2

Subtract $3$ from $10$ .

$d = 7$

$c = 12$

$b = - \frac{7}{2}$

$a = 5$

$d = 7$

$c = 12$

$b = - \frac{7}{2}$

$a = 5$

$d = 7$

$c = 12$

$b = - \frac{7}{2}$

$a = 5$

Step 2.3

List all of the solutions.

$d = 7, c = 12, b = - \frac{7}{2}, a = 5$