Linear Algebra Examples

$[\begin{array}{c} y & 4 \\ 3 & 2 x \end{array}] - [\begin{array}{c} x + 3 & - 3 \\ 2 & 3 y + 1 \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1

Simplify.

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Step 1.1

Subtract the corresponding elements.

$[\begin{array}{c} y - (x + 3) & 4 + 3 \\ 3 - 2 & 2 x - (3 y + 1) \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2

Simplify each element.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Apply the distributive property.

$[\begin{array}{c} y - x - 1 \cdot 3 & 4 + 3 \\ 3 - 2 & 2 x - (3 y + 1) \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2.1.2

Multiply $- 1$ by $3$ .

$[\begin{array}{c} y - x - 3 & 4 + 3 \\ 3 - 2 & 2 x - (3 y + 1) \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2.2

Add $4$ and $3$ .

$[\begin{array}{c} y - x - 3 & 7 \\ 3 - 2 & 2 x - (3 y + 1) \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2.3

Subtract $2$ from $3$ .

$[\begin{array}{c} y - x - 3 & 7 \\ 1 & 2 x - (3 y + 1) \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2.4

Simplify each term.

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Step 1.2.4.1

Apply the distributive property.

$[\begin{array}{c} y - x - 3 & 7 \\ 1 & 2 x - (3 y) - 1 \cdot 1 \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2.4.2

Multiply $3$ by $- 1$ .

$[\begin{array}{c} y - x - 3 & 7 \\ 1 & 2 x - 3 y - 1 \cdot 1 \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 1.2.4.3

Multiply $- 1$ by $1$ .

$[\begin{array}{c} y - x - 3 & 7 \\ 1 & 2 x - 3 y - 1 \end{array}] = [\begin{array}{c} 1 & 7 \\ 1 & - 9 \end{array}]$

Step 2

Find the function rule.

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Step 2.1

Check if the function rule is linear.

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Step 2.1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 2.1.2

Build a set of equations from the table such that $y = a x + b$ .

$7 = a (7) + b 1 = a (1) + b$

Step 2.1.3

Calculate the values of $a$ and $b$ .

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Step 2.1.3.1

Solve for $a$ in $1 = a + b$ .

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Step 2.1.3.1.1

Rewrite the equation as $a + b = 1$ .

$a + b = 1$

$7 = a (7) + b$

Step 2.1.3.1.2

Subtract $b$ from both sides of the equation.

$a = 1 - b$

$7 = a (7) + b$

$a = 1 - b$

$7 = a (7) + b$

Step 2.1.3.2

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

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Step 2.1.3.2.1

Replace all occurrences of $a$ in $7 = a (7) + b$ with $1 - b$ .

$7 = (1 - b) (7) + b$

$a = 1 - b$

Step 2.1.3.2.2

Simplify the right side.

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Step 2.1.3.2.2.1

Simplify $(1 - b) (7) + b$ .

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Step 2.1.3.2.2.1.1

Simplify each term.

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Step 2.1.3.2.2.1.1.1

Apply the distributive property.

$7 = 1 \cdot 7 - b \cdot 7 + b$

$a = 1 - b$

Step 2.1.3.2.2.1.1.2

Multiply $7$ by $1$ .

$7 = 7 - b \cdot 7 + b$

$a = 1 - b$

Step 2.1.3.2.2.1.1.3

Multiply $7$ by $- 1$ .

$7 = 7 - 7 b + b$

$a = 1 - b$

$7 = 7 - 7 b + b$

$a = 1 - b$

Step 2.1.3.2.2.1.2

Add $- 7 b$ and $b$ .

$7 = 7 - 6 b$

$a = 1 - b$

$7 = 7 - 6 b$

$a = 1 - b$

$7 = 7 - 6 b$

$a = 1 - b$

$7 = 7 - 6 b$

$a = 1 - b$

Step 2.1.3.3

Solve for $b$ in $7 = 7 - 6 b$ .

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Step 2.1.3.3.1

Rewrite the equation as $7 - 6 b = 7$ .

$7 - 6 b = 7$

$a = 1 - b$

Step 2.1.3.3.2

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

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Step 2.1.3.3.2.1

Subtract $7$ from both sides of the equation.

$- 6 b = 7 - 7$

$a = 1 - b$

Step 2.1.3.3.2.2

Subtract $7$ from $7$ .

$- 6 b = 0$

$a = 1 - b$

$- 6 b = 0$

$a = 1 - b$

Step 2.1.3.3.3

Divide each term in $- 6 b = 0$ by $- 6$ and simplify.

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Step 2.1.3.3.3.1

Divide each term in $- 6 b = 0$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{0}{- 6}$

$a = 1 - b$

Step 2.1.3.3.3.2

Simplify the left side.

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Step 2.1.3.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 2.1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{0}{- 6}$

$a = 1 - b$

Step 2.1.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{0}{- 6}$

$a = 1 - b$

$b = \frac{0}{- 6}$

$a = 1 - b$

$b = \frac{0}{- 6}$

$a = 1 - b$

Step 2.1.3.3.3.3

Simplify the right side.

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Step 2.1.3.3.3.3.1

Divide $0$ by $- 6$ .

$b = 0$

$a = 1 - b$

$b = 0$

$a = 1 - b$

$b = 0$

$a = 1 - b$

$b = 0$

$a = 1 - b$

Step 2.1.3.4

Replace all occurrences of $b$ with $0$ in each equation.

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Step 2.1.3.4.1

Replace all occurrences of $b$ in $a = 1 - b$ with $0$ .

$a = 1 - (0)$

$b = 0$

Step 2.1.3.4.2

Simplify the right side.

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Step 2.1.3.4.2.1

Subtract $0$ from $1$ .

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

Step 2.1.3.5

List all of the solutions.

$a = 1, b = 0$

Step 2.1.4

Calculate the value of $y$ using each $x$ value in the relation and compare this value to the given $y$ value in the relation.

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Step 2.1.4.1

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = 7$ .

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Step 2.1.4.1.1

Multiply $7$ by $1$ .

$y = 7 + 0$

Step 2.1.4.1.2

Add $7$ and $0$ .

$y = 7$

Step 2.1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 7$ . This check passes since $y = 7$ and $y = 7$ .

$7 = 7$

Step 2.1.4.3

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = 1$ .

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Step 2.1.4.3.1

Multiply $1$ by $1$ .

$y = 1 + 0$

Step 2.1.4.3.2

Add $1$ and $0$ .

$y = 1$

Step 2.1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 1$ and $y = 1$ .

$1 = 1$

Step 2.1.4.5

Since $y = y$ for the corresponding $x$ values, the function is linear.

The function is linear

Step 2.2

Since all $y = y$ , the function is linear and follows the form $y = x$ .

$y = x$

Step 3

Find $y - x - 3$ .

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Step 3.1

Use the function rule equation to find $y - x - 3$ .

$1 = y - x - 3$

Step 3.2

Rewrite the equation as $y - x - 3 = 1$ .

$y - x - 3 = 1$

Step 3.3

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

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Step 3.3.1

Add $x$ to both sides of the equation.

$y - 3 = 1 + x$

Step 3.3.2

Add $3$ to both sides of the equation.

$y = 1 + x + 3$

Step 3.3.3

Add $1$ and $3$ .

$y = x + 4$

Step 4

Find $2 x - 3 y - 1$ .

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Step 4.1

Use the function rule equation to find $2 x - 3 y - 1$ .

$- 9 = 2 x - 3 y - 1$

Step 4.2

Rewrite the equation as $2 x - 3 y - 1 = - 9$ .

$2 x - 3 y - 1 = - 9$

Step 4.3

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

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Step 4.3.1

Add $3 y$ to both sides of the equation.

$2 x - 1 = - 9 + 3 y$

Step 4.3.2

Add $1$ to both sides of the equation.

$2 x = - 9 + 3 y + 1$

Step 4.3.3

Add $- 9$ and $1$ .

$2 x = 3 y - 8$

Step 4.4

Divide each term in $2 x = 3 y - 8$ by $2$ and simplify.

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Step 4.4.1

Divide each term in $2 x = 3 y - 8$ by $2$ .

$\frac{2 x}{2} = \frac{3 y}{2} + \frac{- 8}{2}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3 y}{2} + \frac{- 8}{2}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 y}{2} + \frac{- 8}{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $- 8$ by $2$ .

$x = \frac{3 y}{2} - 4$

Step 5

List all of the solutions.

$y = x + 4 x = \frac{3 y}{2} - 4$