Linear Algebra Examples

$[\begin{array}{c} 2 & 3 \\ 4 & 7 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 1 \\ 1 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 & 3 \\ 4 & 7 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 2 x + 3 y \\ 4 x + 7 y \end{array}] = [\begin{array}{c} 1 \\ 1 \end{array}]$

Step 2

Write as a linear system of equations.

$2 x + 3 y = 1$

$4 x + 7 y = 1$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $x$ in $2 x + 3 y = 1$ .

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Step 3.1.1

Subtract $3 y$ from both sides of the equation.

$2 x = 1 - 3 y$

$4 x + 7 y = 1$

Step 3.1.2

Divide each term in $2 x = 1 - 3 y$ by $2$ and simplify.

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Step 3.1.2.1

Divide each term in $2 x = 1 - 3 y$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2} + \frac{- 3 y}{2}$

$4 x + 7 y = 1$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2} + \frac{- 3 y}{2}$

$4 x + 7 y = 1$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2} + \frac{- 3 y}{2}$

$4 x + 7 y = 1$

$x = \frac{1}{2} + \frac{- 3 y}{2}$

$4 x + 7 y = 1$

$x = \frac{1}{2} + \frac{- 3 y}{2}$

$4 x + 7 y = 1$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative in front of the fraction.

$x = \frac{1}{2} - \frac{3 y}{2}$

$4 x + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$4 x + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$4 x + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$4 x + 7 y = 1$

Step 3.2

Replace all occurrences of $x$ with $\frac{1}{2} - \frac{3 y}{2}$ in each equation.

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Step 3.2.1

Replace all occurrences of $x$ in $4 x + 7 y = 1$ with $\frac{1}{2} - \frac{3 y}{2}$ .

$4 (\frac{1}{2} - \frac{3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $4 (\frac{1}{2} - \frac{3 y}{2}) + 7 y$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$4 (\frac{1}{2}) + 4 (- \frac{3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.1

Factor $2$ out of $4$ .

$2 (2) (\frac{1}{2}) + 4 (- \frac{3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.2.2

Cancel the common factor.

$2 \cdot (2 (\frac{1}{2})) + 4 (- \frac{3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.2.3

Rewrite the expression.

$2 + 4 (- \frac{3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$2 + 4 (- \frac{3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.3.1

Move the leading negative in $- \frac{3 y}{2}$ into the numerator.

$2 + 4 (\frac{- 3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3.2

Factor $2$ out of $4$ .

$2 + 2 (2) (\frac{- 3 y}{2}) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3.3

Cancel the common factor.

$2 + 2 \cdot (2 (\frac{- 3 y}{2})) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.3.4

Rewrite the expression.

$2 + 2 (- 3 y) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$2 + 2 (- 3 y) + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.1.4

Multiply $- 3$ by $2$ .

$2 - 6 y + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$2 - 6 y + 7 y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.2.2.1.2

Add $- 6 y$ and $7 y$ .

$2 + y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$2 + y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$2 + y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$2 + y = 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.3

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

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Step 3.3.1

Subtract $2$ from both sides of the equation.

$y = 1 - 2$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.3.2

Subtract $2$ from $1$ .

$y = - 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

$y = - 1$

$x = \frac{1}{2} - \frac{3 y}{2}$

Step 3.4

Replace all occurrences of $y$ with $- 1$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = \frac{1}{2} - \frac{3 y}{2}$ with $- 1$ .

$x = \frac{1}{2} - \frac{3 (- 1)}{2}$

$y = - 1$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{1}{2} - \frac{3 (- 1)}{2}$ .

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Step 3.4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{1 - 3 \cdot - 1}{2}$

$y = - 1$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Multiply $- 3$ by $- 1$ .

$x = \frac{1 + 3}{2}$

$y = - 1$

Step 3.4.2.1.2.2

Add $1$ and $3$ .

$x = \frac{4}{2}$

$y = - 1$

Step 3.4.2.1.2.3

Divide $4$ by $2$ .

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

Step 3.5

List all of the solutions.

$x = 2, y = - 1$

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