Linear Algebra Examples

- 7 y^{2} + z y - x = 0

$- 7 y^{2} + z y - x = 0$

Step 1

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values

a = - 7

$a = - 7$ ,

b = z

$b = z$ , and

c = - x

$c = - x$ into the quadratic formula and solve for

y

$y$ .

\frac{- z \pm \sqrt{z^{2} - 4 \cdot (- 7 \cdot (- x))}}{2 \cdot - 7}

$\frac{- z \pm \sqrt{z^{2} - 4 \cdot (- 7 \cdot (- x))}}{2 \cdot - 7}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply

- 4 \cdot - 7 \cdot - 1

$- 4 \cdot - 7 \cdot - 1$ .

Tap for more steps...

Step 3.1.1

Multiply

- 4

$- 4$ by

- 7

$- 7$ .

y = \frac{- z \pm \sqrt{z^{2} + 28 \cdot (- 1 x)}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} + 28 \cdot (- 1 x)}}{2 \cdot - 7}$

Step 3.1.2

Multiply

28

$28$ by

- 1

$- 1$ .

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}$

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}$

Step 3.2

Multiply

2

$2$ by

- 7

$- 7$ .

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}$

Step 3.3

Simplify

\frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}

$\frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}$ .

y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}$

y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}$

Step 4

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

Tap for more steps...

Step 4.1

Multiply

- 4 \cdot - 7 \cdot - 1

$- 4 \cdot - 7 \cdot - 1$ .

Tap for more steps...

Step 4.1.1

Multiply

- 4

$- 4$ by

- 7

$- 7$ .

y = \frac{- z \pm \sqrt{z^{2} + 28 \cdot (- 1 x)}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} + 28 \cdot (- 1 x)}}{2 \cdot - 7}$

Step 4.1.2

Multiply

28

$28$ by

- 1

$- 1$ .

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}$

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}$

Step 4.2

Multiply

2

$2$ by

- 7

$- 7$ .

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}$

Step 4.3

Simplify

\frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}

$\frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}$ .

y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}$

Step 4.4

Change the

\pm

$\pm$ to

+

$+$ .

y = \frac{z + \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z + \sqrt{z^{2} - 28 x}}{14}$

y = \frac{z + \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z + \sqrt{z^{2} - 28 x}}{14}$

Step 5

Simplify the expression to solve for the

-

$-$ portion of the

\pm

$\pm$ .

Tap for more steps...

Step 5.1

Multiply

- 4 \cdot - 7 \cdot - 1

$- 4 \cdot - 7 \cdot - 1$ .

Tap for more steps...

Step 5.1.1

Multiply

- 4

$- 4$ by

- 7

$- 7$ .

y = \frac{- z \pm \sqrt{z^{2} + 28 \cdot (- 1 x)}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} + 28 \cdot (- 1 x)}}{2 \cdot - 7}$

Step 5.1.2

Multiply

28

$28$ by

- 1

$- 1$ .

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}$

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{2 \cdot - 7}$

Step 5.2

Multiply

2

$2$ by

- 7

$- 7$ .

y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}

$y = \frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}$

Step 5.3

Simplify

\frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}

$\frac{- z \pm \sqrt{z^{2} - 28 x}}{- 14}$ .

y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z \pm \sqrt{z^{2} - 28 x}}{14}$

Step 5.4

Change the

\pm

$\pm$ to

-

$-$ .

y = \frac{z - \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z - \sqrt{z^{2} - 28 x}}{14}$

y = \frac{z - \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z - \sqrt{z^{2} - 28 x}}{14}$

Step 6

The final answer is the combination of both solutions.

y = \frac{z + \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z + \sqrt{z^{2} - 28 x}}{14}$

y = \frac{z - \sqrt{z^{2} - 28 x}}{14}

$y = \frac{z - \sqrt{z^{2} - 28 x}}{14}$

Step 7

Set the radicand in

\sqrt{z^{2} - 28 x}

$\sqrt{z^{2} - 28 x}$ greater than or equal to

0

$0$ to find where the expression is defined.

z^{2} - 28 x \geq 0

$z^{2} - 28 x \geq 0$

Step 8

Solve for

z

$z$ .

Tap for more steps...

Step 8.1

Add

28 x

$28 x$ to both sides of the inequality.

z^{2} \geq 28 x

$z^{2} \geq 28 x$

Step 8.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

\sqrt{z^{2}} \geq \sqrt{28 x}

$\sqrt{z^{2}} \geq \sqrt{28 x}$

Step 8.3

Simplify the equation.

Tap for more steps...

Step 8.3.1

Simplify the left side.

Tap for more steps...

Step 8.3.1.1

Pull terms out from under the radical.

| z | \geq \sqrt{28 x}

$| z | \geq \sqrt{28 x}$

| z | \geq \sqrt{28 x}

$| z | \geq \sqrt{28 x}$

Step 8.3.2

Simplify the right side.

Tap for more steps...

Step 8.3.2.1

Simplify

\sqrt{28 x}

$\sqrt{28 x}$ .

Tap for more steps...

Step 8.3.2.1.1

Rewrite

28 x

$28 x$ as

2^{2} \cdot (7 x)

$2^{2} \cdot (7 x)$ .

Tap for more steps...

Step 8.3.2.1.1.1

Factor

4

$4$ out of

28

$28$ .

| z | \geq \sqrt{4 (7) x}

$| z | \geq \sqrt{4 (7) x}$

Step 8.3.2.1.1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

| z | \geq \sqrt{2^{2} \cdot 7 x}

$| z | \geq \sqrt{2^{2} \cdot 7 x}$

Step 8.3.2.1.1.3

Add parentheses.

| z | \geq \sqrt{2^{2} \cdot (7 x)}

$| z | \geq \sqrt{2^{2} \cdot (7 x)}$

| z | \geq \sqrt{2^{2} \cdot (7 x)}

$| z | \geq \sqrt{2^{2} \cdot (7 x)}$

Step 8.3.2.1.2

Pull terms out from under the radical.

| z | \geq | 2 | \sqrt{7 x}

$| z | \geq | 2 | \sqrt{7 x}$

Step 8.3.2.1.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

2

$2$ is

2

$2$ .

| z | \geq 2 \sqrt{7 x}

$| z | \geq 2 \sqrt{7 x}$

| z | \geq 2 \sqrt{7 x}

$| z | \geq 2 \sqrt{7 x}$

| z | \geq 2 \sqrt{7 x}

$| z | \geq 2 \sqrt{7 x}$

| z | \geq 2 \sqrt{7 x}

$| z | \geq 2 \sqrt{7 x}$

Step 8.4

Write

| z | \geq 2 \sqrt{7 x}

$| z | \geq 2 \sqrt{7 x}$ as a piecewise.

Tap for more steps...

Step 8.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

z \geq 0

$z \geq 0$

Step 8.4.2

In the piece where

z

$z$ is non-negative, remove the absolute value.

z \geq 2 \sqrt{7 x}

$z \geq 2 \sqrt{7 x}$

Step 8.4.3

Find the domain of

z \geq 2 \sqrt{7 x}

$z \geq 2 \sqrt{7 x}$ and find the intersection with

z \geq 0

$z \geq 0$ .

Tap for more steps...

Step 8.4.3.1

Find the domain of

z \geq 2 \sqrt{7 x}

$z \geq 2 \sqrt{7 x}$ .

Tap for more steps...

Step 8.4.3.1.1

Set the radicand in

\sqrt{7 x}

$\sqrt{7 x}$ greater than or equal to

0

$0$ to find where the expression is defined.

7 x \geq 0

$7 x \geq 0$

Step 8.4.3.1.2

Divide each term in

7 x \geq 0

$7 x \geq 0$ by

7

$7$ and simplify.

Tap for more steps...

Step 8.4.3.1.2.1

Divide each term in

7 x \geq 0

$7 x \geq 0$ by

7

$7$ .

\frac{7 x}{7} \geq \frac{0}{7}

$\frac{7 x}{7} \geq \frac{0}{7}$

Step 8.4.3.1.2.2

Simplify the left side.

Tap for more steps...

Step 8.4.3.1.2.2.1

Cancel the common factor of

7

$7$ .

Tap for more steps...

Step 8.4.3.1.2.2.1.1

Cancel the common factor.

\frac{7 x}{7} \geq \frac{0}{7}

$\frac{7 x}{7} \geq \frac{0}{7}$

Step 8.4.3.1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x \geq \frac{0}{7}

$x \geq \frac{0}{7}$

x \geq \frac{0}{7}

$x \geq \frac{0}{7}$

x \geq \frac{0}{7}

$x \geq \frac{0}{7}$

Step 8.4.3.1.2.3

Simplify the right side.

Tap for more steps...

Step 8.4.3.1.2.3.1

Divide

0

$0$ by

7

$7$ .

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

Step 8.4.3.1.3

The domain is all values of

z

$z$ that make the expression defined.

[0, \infty)

$[0, \infty)$

[0, \infty)

$[0, \infty)$

Step 8.4.3.2

Find the intersection of

z \geq 0

$z \geq 0$ and

[0, \infty)

$[0, \infty)$ .

z \geq 0

$z \geq 0$

z \geq 0

$z \geq 0$

Step 8.4.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

z < 0

$z < 0$

Step 8.4.5

In the piece where

z

$z$ is negative, remove the absolute value and multiply by

- 1

$- 1$ .

- z \geq 2 \sqrt{7 x}

$- z \geq 2 \sqrt{7 x}$

Step 8.4.6

Find the domain of

- z \geq 2 \sqrt{7 x}

$- z \geq 2 \sqrt{7 x}$ and find the intersection with

z < 0

$z < 0$ .

Tap for more steps...

Step 8.4.6.1

Find the domain of

- z \geq 2 \sqrt{7 x}

$- z \geq 2 \sqrt{7 x}$ .

Tap for more steps...

Step 8.4.6.1.1

Set the radicand in

\sqrt{7 x}

$\sqrt{7 x}$ greater than or equal to

0

$0$ to find where the expression is defined.

7 x \geq 0

$7 x \geq 0$

Step 8.4.6.1.2

Divide each term in

7 x \geq 0

$7 x \geq 0$ by

7

$7$ and simplify.

Tap for more steps...

Step 8.4.6.1.2.1

Divide each term in

7 x \geq 0

$7 x \geq 0$ by

7

$7$ .

\frac{7 x}{7} \geq \frac{0}{7}

$\frac{7 x}{7} \geq \frac{0}{7}$

Step 8.4.6.1.2.2

Simplify the left side.

Tap for more steps...

Step 8.4.6.1.2.2.1

Cancel the common factor of

7

$7$ .

Tap for more steps...

Step 8.4.6.1.2.2.1.1

Cancel the common factor.

\frac{7 x}{7} \geq \frac{0}{7}

$\frac{7 x}{7} \geq \frac{0}{7}$

Step 8.4.6.1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x \geq \frac{0}{7}

$x \geq \frac{0}{7}$

x \geq \frac{0}{7}

$x \geq \frac{0}{7}$

x \geq \frac{0}{7}

$x \geq \frac{0}{7}$

Step 8.4.6.1.2.3

Simplify the right side.

Tap for more steps...

Step 8.4.6.1.2.3.1

Divide

0

$0$ by

7

$7$ .

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

x \geq 0

$x \geq 0$

Step 8.4.6.1.3

The domain is all values of

z

$z$ that make the expression defined.

[0, \infty)

$[0, \infty)$

[0, \infty)

$[0, \infty)$

Step 8.4.6.2

Find the intersection of

z < 0

$z < 0$ and

[0, \infty)

$[0, \infty)$ .

No solution

$No solution$

No solution

$No solution$

Step 8.4.7

Write as a piecewise.

{\begin{matrix} z \geq 2 \sqrt{7 x} & z \geq 0 \end{matrix}

${\begin{cases} z \geq 2 \sqrt{7 x} & z \geq 0 \end{cases}$

{\begin{matrix} z \geq 2 \sqrt{7 x} & z \geq 0 \end{matrix}

${\begin{cases} z \geq 2 \sqrt{7 x} & z \geq 0 \end{cases}$

Step 8.5

Find the intersection of

z \geq 2 \sqrt{7 x}

$z \geq 2 \sqrt{7 x}$ and

z \geq 0

$z \geq 0$ .

z \geq 2 \sqrt{7 x}

$z \geq 2 \sqrt{7 x}$ and

z \geq 0

$z \geq 0$

Step 8.6

Find the union of the solutions.

z \geq N o (Max i m u m)

$z \geq N o (Max i m u m)$

z \geq N o (Max i m u m)

$z \geq N o (Max i m u m)$

Step 9

The domain is all real numbers.

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$

Set-Builder Notation:

${z | z \in ℝ}$

Step 10