Answer :
Let's analyze each statement one by one.
### Statement A
The equation represents a proportional relationship.
For a linear equation [tex]\( y = mx \)[/tex] to represent a proportional relationship, the equation must pass through the origin (0, 0). The given equation is [tex]\( y = 1.4x \)[/tex], which is of the form [tex]\( y = mx \)[/tex]. Since there is no constant term (the y-intercept is 0), the line passes through the origin. Therefore, this statement is true.
### Statement B
The unit rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is 1.4.
The unit rate of change in a linear equation [tex]\( y = mx \)[/tex] is represented by the slope [tex]\( m \)[/tex]. Here, the slope [tex]\( m \)[/tex] is 1.4, which means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1.4 units. Therefore, this statement is true.
### Statement C
The slope of the line is [tex]\(\frac{5}{7}\)[/tex].
The given equation has a slope of 1.4. To verify whether this is equivalent to [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[\frac{5}{7} \approx 0.714 \][/tex]
Since 1.4 is not equal to 0.714, the slope is not [tex]\(\frac{5}{7}\)[/tex]. Therefore, this statement is false.
### Statement D
A change of 2 units in [tex]\( x \)[/tex] results in a change of 2.8 units in [tex]\( y \)[/tex].
To find the change in [tex]\( y \)[/tex] for a change in [tex]\( x \)[/tex] of 2 units, we use the slope:
[tex]\[ \text{Change in } y = \text{slope} \times \text{change in } x = 1.4 \times 2 = 2.8 \][/tex]
Thus, a 2-unit change in [tex]\( x \)[/tex] results in a 2.8-unit change in [tex]\( y \)[/tex]. Therefore, this statement is true.
### Statement E
A change of 3 units in [tex]\( x \)[/tex] results in a change of 1.2 units in [tex]\( y \)[/tex].
To find the change in [tex]\( y \)[/tex] for a change in [tex]\( x \)[/tex] of 3 units, we again use the slope:
[tex]\[ \text{Change in } y = \text{slope} \times \text{change in } x = 1.4 \times 3 = 4.2 \][/tex]
Thus, a 3-unit change in [tex]\( x \)[/tex] results in a 4.2-unit change in [tex]\( y \)[/tex], not 1.2 units. Therefore, this statement is false.
### Summary
The true statements are:
- A. The equation represents a proportional relationship.
- B. The unit rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is 1.4.
- D. A change of 2 units in [tex]\( x \)[/tex] results in a change of 2.8 units in [tex]\( y \)[/tex].
So the correct answers are A, B, and D.
### Statement A
The equation represents a proportional relationship.
For a linear equation [tex]\( y = mx \)[/tex] to represent a proportional relationship, the equation must pass through the origin (0, 0). The given equation is [tex]\( y = 1.4x \)[/tex], which is of the form [tex]\( y = mx \)[/tex]. Since there is no constant term (the y-intercept is 0), the line passes through the origin. Therefore, this statement is true.
### Statement B
The unit rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is 1.4.
The unit rate of change in a linear equation [tex]\( y = mx \)[/tex] is represented by the slope [tex]\( m \)[/tex]. Here, the slope [tex]\( m \)[/tex] is 1.4, which means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1.4 units. Therefore, this statement is true.
### Statement C
The slope of the line is [tex]\(\frac{5}{7}\)[/tex].
The given equation has a slope of 1.4. To verify whether this is equivalent to [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[\frac{5}{7} \approx 0.714 \][/tex]
Since 1.4 is not equal to 0.714, the slope is not [tex]\(\frac{5}{7}\)[/tex]. Therefore, this statement is false.
### Statement D
A change of 2 units in [tex]\( x \)[/tex] results in a change of 2.8 units in [tex]\( y \)[/tex].
To find the change in [tex]\( y \)[/tex] for a change in [tex]\( x \)[/tex] of 2 units, we use the slope:
[tex]\[ \text{Change in } y = \text{slope} \times \text{change in } x = 1.4 \times 2 = 2.8 \][/tex]
Thus, a 2-unit change in [tex]\( x \)[/tex] results in a 2.8-unit change in [tex]\( y \)[/tex]. Therefore, this statement is true.
### Statement E
A change of 3 units in [tex]\( x \)[/tex] results in a change of 1.2 units in [tex]\( y \)[/tex].
To find the change in [tex]\( y \)[/tex] for a change in [tex]\( x \)[/tex] of 3 units, we again use the slope:
[tex]\[ \text{Change in } y = \text{slope} \times \text{change in } x = 1.4 \times 3 = 4.2 \][/tex]
Thus, a 3-unit change in [tex]\( x \)[/tex] results in a 4.2-unit change in [tex]\( y \)[/tex], not 1.2 units. Therefore, this statement is false.
### Summary
The true statements are:
- A. The equation represents a proportional relationship.
- B. The unit rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is 1.4.
- D. A change of 2 units in [tex]\( x \)[/tex] results in a change of 2.8 units in [tex]\( y \)[/tex].
So the correct answers are A, B, and D.
