Linear Algebra Examples

$[\begin{array}{c} x - 1 & 4 \\ y + 3 & - 7 \end{array}] = [\begin{array}{c} 0 & 4 \\ - 2 & - 7 \end{array}]$

Step 1

Find the function rule.

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Step 1.1

Check if the function rule is linear.

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Step 1.1.1

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 1.1.2

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

$4 = a (4) + b - 7 = a (- 7) + b$

Step 1.1.3

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

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Step 1.1.3.1

Solve for $b$ in $4 = a (4) + b$ .

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Step 1.1.3.1.1

Rewrite the equation as $a (4) + b = 4$ .

$a (4) + b = 4$

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

Step 1.1.3.1.2

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

$4 a + b = 4$

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

Step 1.1.3.1.3

Subtract $4 a$ from both sides of the equation.

$b = 4 - 4 a$

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

$b = 4 - 4 a$

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

Step 1.1.3.2

Replace all occurrences of $b$ with $4 - 4 a$ in each equation.

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Step 1.1.3.2.1

Replace all occurrences of $b$ in $- 7 = a (- 7) + b$ with $4 - 4 a$ .

$- 7 = a (- 7) + 4 - 4 a$

$b = 4 - 4 a$

Step 1.1.3.2.2

Simplify $- 7 = a (- 7) + 4 - 4 a$ .

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Step 1.1.3.2.2.1

Simplify the left side.

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Step 1.1.3.2.2.1.1

Remove parentheses.

$- 7 = a (- 7) + 4 - 4 a$

$b = 4 - 4 a$

$- 7 = a (- 7) + 4 - 4 a$

$b = 4 - 4 a$

Step 1.1.3.2.2.2

Simplify the right side.

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Step 1.1.3.2.2.2.1

Simplify $a (- 7) + 4 - 4 a$ .

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Step 1.1.3.2.2.2.1.1

Move $- 7$ to the left of $a$ .

$- 7 = - 7 a + 4 - 4 a$

$b = 4 - 4 a$

Step 1.1.3.2.2.2.1.2

Subtract $4 a$ from $- 7 a$ .

$- 7 = - 11 a + 4$

$b = 4 - 4 a$

$- 7 = - 11 a + 4$

$b = 4 - 4 a$

$- 7 = - 11 a + 4$

$b = 4 - 4 a$

$- 7 = - 11 a + 4$

$b = 4 - 4 a$

$- 7 = - 11 a + 4$

$b = 4 - 4 a$

Step 1.1.3.3

Solve for $a$ in $- 7 = - 11 a + 4$ .

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Step 1.1.3.3.1

Rewrite the equation as $- 11 a + 4 = - 7$ .

$- 11 a + 4 = - 7$

$b = 4 - 4 a$

Step 1.1.3.3.2

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

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Step 1.1.3.3.2.1

Subtract $4$ from both sides of the equation.

$- 11 a = - 7 - 4$

$b = 4 - 4 a$

Step 1.1.3.3.2.2

Subtract $4$ from $- 7$ .

$- 11 a = - 11$

$b = 4 - 4 a$

$- 11 a = - 11$

$b = 4 - 4 a$

Step 1.1.3.3.3

Divide each term in $- 11 a = - 11$ by $- 11$ and simplify.

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Step 1.1.3.3.3.1

Divide each term in $- 11 a = - 11$ by $- 11$ .

$\frac{- 11 a}{- 11} = \frac{- 11}{- 11}$

$b = 4 - 4 a$

Step 1.1.3.3.3.2

Simplify the left side.

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Step 1.1.3.3.3.2.1

Cancel the common factor of $- 11$ .

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Step 1.1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 11 a}{- 11} = \frac{- 11}{- 11}$

$b = 4 - 4 a$

Step 1.1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 11}{- 11}$

$b = 4 - 4 a$

$a = \frac{- 11}{- 11}$

$b = 4 - 4 a$

$a = \frac{- 11}{- 11}$

$b = 4 - 4 a$

Step 1.1.3.3.3.3

Simplify the right side.

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Step 1.1.3.3.3.3.1

Divide $- 11$ by $- 11$ .

$a = 1$

$b = 4 - 4 a$

$a = 1$

$b = 4 - 4 a$

$a = 1$

$b = 4 - 4 a$

$a = 1$

$b = 4 - 4 a$

Step 1.1.3.4

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

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Step 1.1.3.4.1

Replace all occurrences of $a$ in $b = 4 - 4 a$ with $1$ .

$b = 4 - 4 \cdot 1$

$a = 1$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Simplify $4 - 4 \cdot 1$ .

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Step 1.1.3.4.2.1.1

Multiply $- 4$ by $1$ .

$b = 4 - 4$

$a = 1$

Step 1.1.3.4.2.1.2

Subtract $4$ from $4$ .

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

Step 1.1.3.5

List all of the solutions.

$b = 0, a = 1$

Step 1.1.4

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 1.1.4.1

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = 4$ .

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Step 1.1.4.1.1

Multiply $4$ by $1$ .

$y = 4 + 0$

Step 1.1.4.1.2

Add $4$ and $0$ .

$y = 4$

Step 1.1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 4$ and $y = 4$ .

$4 = 4$

Step 1.1.4.3

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

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Step 1.1.4.3.1

Multiply $- 7$ by $1$ .

$y = - 7 + 0$

Step 1.1.4.3.2

Add $- 7$ and $0$ .

$y = - 7$

Step 1.1.4.4

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 1.1.4.5

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

The function is linear

Step 1.2

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

$y = x$

Step 2

Find $x - 1$ .

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Step 2.1

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

$0 = x - 1$

Step 2.2

Rewrite the equation as $x - 1 = 0$ .

$x - 1 = 0$

Step 2.3

Add $1$ to both sides of the equation.

$x = 1$

Step 3

Find $y + 3$ .

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Step 3.1

Use the function rule equation to find $y + 3$ .

$- 2 = y + 3$

Step 3.2

Rewrite the equation as $y + 3 = - 2$ .

$y + 3 = - 2$

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 $3$ from both sides of the equation.

$y = - 2 - 3$

Step 3.3.2

Subtract $3$ from $- 2$ .

$y = - 5$

Step 4

List all of the solutions.

$x = 1 y = - 5$