determine whether the given equation is symmetric with respect to the x - axis, the y - axis, and the origin. (use algebra)
1. $y = |x + 5|$
2. $2x - 5 = 3y$
3. $6x + 7y = 0$
4. $2y^{2}=5x^{2}+12$
5. $y=\frac{-4}{x}$

Question

Accepted Answer

### Problem 1: $ y = |x + 5| $
#### Explanation:
- **x - axis symmetry**: Replace $ y $ with $ -y $. We get $ -y = |x + 5| $, or $ y = -|x + 5| $, which is not the same as the original equation. So, not symmetric about the x - axis.
- **y - axis symmetry**: Replace $ x $ with $ -x $. We get $ y = |-x + 5|=|x - 5| $, which is not the same as $ y = |x + 5| $ (unless $ x = 0 $, but generally, they are different). So, not symmetric about the y - axis.
- **Origin symmetry**: Replace $ x $ with $ -x $ and $ y $ with $ -y $. We get $ -y = |-x + 5| $, or $ y=-|x - 5| $, which is not the same as the original equation. So, not symmetric about the origin.

#### Answer:
Not symmetric about the x - axis, y - axis, or origin.

### Problem 2: $ 2x-5 = 3y $
#### Explanation:
- **x - axis symmetry**: Replace $ y $ with $ -y $. We get $ 2x - 5=-3y $, or $ 3y=5 - 2x $, which is not the same as $ 3y = 2x - 5 $. So, not symmetric about the x - axis.
- **y - axis symmetry**: Replace $ x $ with $ -x $. We get $ -2x-5 = 3y $, or $ 3y=-2x - 5 $, which is not the same as $ 3y = 2x - 5 $. So, not symmetric about the y - axis.
- **Origin symmetry**: Replace $ x $ with $ -x $ and $ y $ with $ -y $. We get $ -2x - 5=-3y $, or $ 3y=2x + 5 $, which is not the same as $ 3y = 2x - 5 $. So, not symmetric about the origin.

#### Answer:
Not symmetric about the x - axis, y - axis, or origin.

### Problem 3: $ 6x + 7y=0 $
#### Explanation:
- **x - axis symmetry**: Replace $ y $ with $ -y $. We get $ 6x-7y = 0 $, which is not the same as $ 6x + 7y=0 $. So, not symmetric about the x - axis.
- **y - axis symmetry**: Replace $ x $ with $ -x $. We get $ -6x + 7y=0 $, or $ 6x-7y = 0 $, which is not the same as $ 6x + 7y=0 $. So, not symmetric about the y - axis.
- **Origin symmetry**: Replace $ x $ with $ -x $ and $ y $ with $ -y $. We get $ -6x-7y = 0 $, or $ 6x + 7y=0 $ (multiply both sides by - 1). So, symmetric about the origin.

#### Answer:
Not symmetric about the x - axis or y - axis, symmetric about the origin.

### Problem 4: $ 2y^{2}=5x^{2}+12 $
#### Explanation:
- **x - axis symmetry**: Replace $ y $ with $ -y $. We get $ 2(-y)^{2}=5x^{2}+12 $, since $ (-y)^{2}=y^{2} $, the equation becomes $ 2y^{2}=5x^{2}+12 $, which is the same as the original equation. So, symmetric about the x - axis.
- **y - axis symmetry**: Replace $ x $ with $ -x $. We get $ 2y^{2}=5(-x)^{2}+12 $, since $ (-x)^{2}=x^{2} $, the equation becomes $ 2y^{2}=5x^{2}+12 $, which is the same as the original equation. So, symmetric about the y - axis.
- **Origin symmetry**: Replace $ x $ with $ -x $ and $ y $ with $ -y $. We get $ 2(-y)^{2}=5(-x)^{2}+12 $, which simplifies to $ 2y^{2}=5x^{2}+12 $, the same as the original equation. So, symmetric about the origin.

#### Answer:
Symmetric about the x - axis, y - axis, and origin.

### Problem 5: $ y=\frac{-4}{x} $
#### Explanation:
- **x - axis symmetry**: Replace $ y $ with $ -y $. We get $ -y=\frac{-4}{x} $, or $ y = \frac{4}{x} $, which is not the same as the original equation $ y=\frac{-4}{x} $. So, not symmetric about the x - axis.
- **y - axis symmetry**: Replace $ x $ with $ -x $. We get $ y=\frac{-4}{-x}=\frac{4}{x} $, which is not the same as the original equation. So, not symmetric about the y - axis.
- **Origin symmetry**: Replace $ x $ with $ -x $ and $ y $ with $ -y $. We get $ -y=\frac{-4}{-x} $, or $ -y=\frac{4}{x} $, or $ y=\frac{-4}{x} $, which is the same as the original equation. So, symmetric about the origin.

#### Answer:
Not symmetric about the x - axis or y - axis, symmetric about the origin.

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( y = |x + 5| \)

Answer:

Problem 2: \( 2x-5 = 3y \)