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To determine whether each equation represents a direct variation, we need to see if they can be written in the form [tex]\( y = kx \)[/tex] where [tex]\( k \)[/tex] is a constant. In a direct variation, [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex].
Let’s analyze each equation one by one:
1. [tex]\( y = x \)[/tex]
- This fits the form [tex]\( y = kx \)[/tex] with [tex]\( k = 1 \)[/tex]. Therefore, this is a direct variation.
2. [tex]\( y = x + 2 \)[/tex]
- This equation cannot be written in the form [tex]\( y = kx \)[/tex] because of the additional [tex]\( +2 \)[/tex] term. Therefore, this is not a direct variation.
3. [tex]\( y = 6 \)[/tex]
- This represents a horizontal line, which means [tex]\( y \)[/tex] is constant and not dependent on [tex]\( x \)[/tex]. Hence, this is not a direct variation.
4. [tex]\( y = 4 - x \)[/tex]
- This can be rearranged to [tex]\( y = -x + 4 \)[/tex], which is a linear equation with a slope of [tex]\(-1\)[/tex] but includes the constant term [tex]\( +4 \)[/tex]. Therefore, this is not a direct variation, since it cannot be expressed purely as [tex]\( y = kx \)[/tex].
5. [tex]\( \frac{y}{x} = 8 \)[/tex]
- Multiplying both sides by [tex]\( x \)[/tex] gives [tex]\( y = 8x \)[/tex]. This is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = 8 \)[/tex]. Therefore, this is a direct variation.
6. [tex]\( y = -7x \)[/tex]
- This can directly be written in the form [tex]\( y = kx \)[/tex] with [tex]\( k = -7 \)[/tex]. Therefore, this is a direct variation.
7. [tex]\( x = -1 \)[/tex]
- This represents a vertical line, which means [tex]\( x \)[/tex] is constant and [tex]\( y \)[/tex] is not defined in terms of [tex]\( x \)[/tex]. Therefore, this is not a direct variation.
8. [tex]\( \frac{y}{2x} = 5 \)[/tex]
- Multiplying both sides by [tex]\( 2x \)[/tex] gives [tex]\( y = 10x \)[/tex]. This is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = 10 \)[/tex]. Therefore, this is a direct variation.
In conclusion:
1. [tex]\( y = x \)[/tex] — Direct Variation
2. [tex]\( y = x + 2 \)[/tex] — Not a Direct Variation
3. [tex]\( y = 6 \)[/tex] — Not a Direct Variation
4. [tex]\( y = 4 - x \)[/tex] — Not a Direct Variation
5. [tex]\( \frac{y}{x} = 8 \)[/tex] — Direct Variation
6. [tex]\( y = -7x \)[/tex] — Direct Variation
7. [tex]\( x = -1 \)[/tex] — Not a Direct Variation
8. [tex]\( \frac{y}{2x} = 5 \)[/tex] — Direct Variation
Let’s analyze each equation one by one:
1. [tex]\( y = x \)[/tex]
- This fits the form [tex]\( y = kx \)[/tex] with [tex]\( k = 1 \)[/tex]. Therefore, this is a direct variation.
2. [tex]\( y = x + 2 \)[/tex]
- This equation cannot be written in the form [tex]\( y = kx \)[/tex] because of the additional [tex]\( +2 \)[/tex] term. Therefore, this is not a direct variation.
3. [tex]\( y = 6 \)[/tex]
- This represents a horizontal line, which means [tex]\( y \)[/tex] is constant and not dependent on [tex]\( x \)[/tex]. Hence, this is not a direct variation.
4. [tex]\( y = 4 - x \)[/tex]
- This can be rearranged to [tex]\( y = -x + 4 \)[/tex], which is a linear equation with a slope of [tex]\(-1\)[/tex] but includes the constant term [tex]\( +4 \)[/tex]. Therefore, this is not a direct variation, since it cannot be expressed purely as [tex]\( y = kx \)[/tex].
5. [tex]\( \frac{y}{x} = 8 \)[/tex]
- Multiplying both sides by [tex]\( x \)[/tex] gives [tex]\( y = 8x \)[/tex]. This is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = 8 \)[/tex]. Therefore, this is a direct variation.
6. [tex]\( y = -7x \)[/tex]
- This can directly be written in the form [tex]\( y = kx \)[/tex] with [tex]\( k = -7 \)[/tex]. Therefore, this is a direct variation.
7. [tex]\( x = -1 \)[/tex]
- This represents a vertical line, which means [tex]\( x \)[/tex] is constant and [tex]\( y \)[/tex] is not defined in terms of [tex]\( x \)[/tex]. Therefore, this is not a direct variation.
8. [tex]\( \frac{y}{2x} = 5 \)[/tex]
- Multiplying both sides by [tex]\( 2x \)[/tex] gives [tex]\( y = 10x \)[/tex]. This is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = 10 \)[/tex]. Therefore, this is a direct variation.
In conclusion:
1. [tex]\( y = x \)[/tex] — Direct Variation
2. [tex]\( y = x + 2 \)[/tex] — Not a Direct Variation
3. [tex]\( y = 6 \)[/tex] — Not a Direct Variation
4. [tex]\( y = 4 - x \)[/tex] — Not a Direct Variation
5. [tex]\( \frac{y}{x} = 8 \)[/tex] — Direct Variation
6. [tex]\( y = -7x \)[/tex] — Direct Variation
7. [tex]\( x = -1 \)[/tex] — Not a Direct Variation
8. [tex]\( \frac{y}{2x} = 5 \)[/tex] — Direct Variation
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