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Sagot :
To analyze the line given by the equation [tex]\( y = -3x + 6 \)[/tex], let's evaluate each statement one by one.
### Statement 1: The line increases from left to right
To determine if the line increases or decreases, we look at the slope. The slope [tex]\( m \)[/tex] in the equation [tex]\( y = mx + b \)[/tex] is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( m = -3 \)[/tex], which is a negative value. A negative slope indicates that the line decreases from left to right, not increases.
Conclusion: This statement is false.
### Statement 2: The line crosses the x-axis at -3
To find the x-intercept of a line, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -3x + 6 \implies 3x = 6 \implies x = 2 \][/tex]
The line crosses the x-axis at [tex]\( x = 2 \)[/tex], not at -3.
Conclusion: This statement is false.
### Statement 3: The line crosses the y-axis at 6
To find the y-intercept, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ y = -3(0) + 6 \implies y = 6 \][/tex]
The line crosses the y-axis at [tex]\( y = 6 \)[/tex], which is correct.
Conclusion: This statement is true.
### Statement 4: As the input increases, the output decreases
As [tex]\( x \)[/tex] (the input) increases, we need to determine what happens to [tex]\( y \)[/tex] (the output). With a negative slope of -3, as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases. This is due to the negative relationship described by the slope.
Conclusion: This statement is true.
### Statement 5: The line does not enter quadrant II at all
Quadrant II is where [tex]\( x \)[/tex] is negative and [tex]\( y \)[/tex] is positive. We need to check if the line ever passes through this quadrant. For [tex]\( x \)[/tex] values less than 0:
[tex]\[ y = -3x + 6 \][/tex]
If [tex]\( x \)[/tex] is negative, then [tex]\( -3x \)[/tex] becomes positive, making [tex]\( y \)[/tex] greater than 6, which is positive. Therefore, the line does enter quadrant II.
Conclusion: This statement is false.
### Statement 6: The point [tex]\((3.5, -4.5)\)[/tex] lies on the line
To verify if the point [tex]\((3.5, -4.5)\)[/tex] lies on the line, we substitute [tex]\( x = 3.5 \)[/tex] into the equation and see if [tex]\( y \)[/tex] equals [tex]\(-4.5\)[/tex]:
[tex]\[ y = -3(3.5) + 6 = -10.5 + 6 = -4.5 \][/tex]
Since the calculated [tex]\( y \)[/tex] value matches the given [tex]\( y \)[/tex] value, the point [tex]\((3.5, -4.5)\)[/tex] lies on the line.
Conclusion: This statement is true.
### Summary
After evaluating each statement, the true statements are:
- The line crosses the y-axis at 6.
- As the input increases, the output decreases.
- The point [tex]\((3.5, -4.5)\)[/tex] lies on the line.
Thus, the true statements are:
[tex]\[ \text{True statements: (c), (d), (f)} \][/tex]
False statements:
- (a) The line increases from left to right.
- (b) The line crosses the [tex]\( x \)[/tex]-axis at -3.
- (e) The line does not enter quadrant II at all.
### Statement 1: The line increases from left to right
To determine if the line increases or decreases, we look at the slope. The slope [tex]\( m \)[/tex] in the equation [tex]\( y = mx + b \)[/tex] is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( m = -3 \)[/tex], which is a negative value. A negative slope indicates that the line decreases from left to right, not increases.
Conclusion: This statement is false.
### Statement 2: The line crosses the x-axis at -3
To find the x-intercept of a line, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -3x + 6 \implies 3x = 6 \implies x = 2 \][/tex]
The line crosses the x-axis at [tex]\( x = 2 \)[/tex], not at -3.
Conclusion: This statement is false.
### Statement 3: The line crosses the y-axis at 6
To find the y-intercept, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ y = -3(0) + 6 \implies y = 6 \][/tex]
The line crosses the y-axis at [tex]\( y = 6 \)[/tex], which is correct.
Conclusion: This statement is true.
### Statement 4: As the input increases, the output decreases
As [tex]\( x \)[/tex] (the input) increases, we need to determine what happens to [tex]\( y \)[/tex] (the output). With a negative slope of -3, as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases. This is due to the negative relationship described by the slope.
Conclusion: This statement is true.
### Statement 5: The line does not enter quadrant II at all
Quadrant II is where [tex]\( x \)[/tex] is negative and [tex]\( y \)[/tex] is positive. We need to check if the line ever passes through this quadrant. For [tex]\( x \)[/tex] values less than 0:
[tex]\[ y = -3x + 6 \][/tex]
If [tex]\( x \)[/tex] is negative, then [tex]\( -3x \)[/tex] becomes positive, making [tex]\( y \)[/tex] greater than 6, which is positive. Therefore, the line does enter quadrant II.
Conclusion: This statement is false.
### Statement 6: The point [tex]\((3.5, -4.5)\)[/tex] lies on the line
To verify if the point [tex]\((3.5, -4.5)\)[/tex] lies on the line, we substitute [tex]\( x = 3.5 \)[/tex] into the equation and see if [tex]\( y \)[/tex] equals [tex]\(-4.5\)[/tex]:
[tex]\[ y = -3(3.5) + 6 = -10.5 + 6 = -4.5 \][/tex]
Since the calculated [tex]\( y \)[/tex] value matches the given [tex]\( y \)[/tex] value, the point [tex]\((3.5, -4.5)\)[/tex] lies on the line.
Conclusion: This statement is true.
### Summary
After evaluating each statement, the true statements are:
- The line crosses the y-axis at 6.
- As the input increases, the output decreases.
- The point [tex]\((3.5, -4.5)\)[/tex] lies on the line.
Thus, the true statements are:
[tex]\[ \text{True statements: (c), (d), (f)} \][/tex]
False statements:
- (a) The line increases from left to right.
- (b) The line crosses the [tex]\( x \)[/tex]-axis at -3.
- (e) The line does not enter quadrant II at all.
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