Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Let's analyze each of the given equations to determine if they could represent the line [tex]\( f(x) = 4x + 3 \)[/tex] that passes through the point [tex]\((1,7)\)[/tex].
### 1. Equation: [tex]\( y - 7 = 3(x - 1) \)[/tex]
First, we need to rewrite this equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]):
1. Distribute the 3 on the right-hand side:
[tex]\[ y - 7 = 3(x - 1) \\ y - 7 = 3x - 3 \][/tex]
2. Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 3 + 7 \\ y = 3x + 4 \][/tex]
The slope ([tex]\( m \)[/tex]) is 3 and the y-intercept is 4. This does not match our original line equation [tex]\( f(x) = 4x + 3 \)[/tex]. So, the first equation does not represent the same line.
### 2. Equation: [tex]\( y - 1 = 3(x - 7) \)[/tex]
Next, we transform this equation to slope-intercept form:
1. Distribute the 3 on the right-hand side:
[tex]\[ y - 1 = 3(x - 7) \\ y - 1 = 3x - 21 \][/tex]
2. Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 21 + 1 \\ y = 3x - 20 \][/tex]
The slope ([tex]\( m \)[/tex]) here is 3 and the y-intercept is -20. This also does not match our original line equation. Thus, the second equation does not represent the same line.
### 3. Equation: [tex]\( y - 7 = 4(x - 1) \)[/tex]
Let’s rewrite this equation in slope-intercept form:
1. Distribute the 4 on the right-hand side:
[tex]\[ y - 7 = 4(x - 1) \\ y - 7 = 4x - 4 \][/tex]
2. Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 4 + 7 \\ y = 4x + 3 \][/tex]
The slope ([tex]\( m \)[/tex]) here is 4 and the y-intercept is 3. This matches our original line equation [tex]\( f(x) = 4x + 3 \)[/tex]. Therefore, the third equation represents the same line.
### 4. Equation: [tex]\( y - 1 = 4(x - 7) \)[/tex]
Finally, transform this to slope-intercept form:
1. Distribute the 4 on the right-hand side:
[tex]\[ y - 1 = 4(x - 7) \\ y - 1 = 4x - 28 \][/tex]
2. Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 28 + 1 \\ y = 4x - 27 \][/tex]
The slope ([tex]\( m \)[/tex]) is 4 but the y-intercept is -27. This does not match our original line's intercept. Thus, the fourth equation does not represent the same line.
### Conclusion
Out of the four given equations, only [tex]\( y - 7 = 4(x - 1) \)[/tex] represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex].
### 1. Equation: [tex]\( y - 7 = 3(x - 1) \)[/tex]
First, we need to rewrite this equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]):
1. Distribute the 3 on the right-hand side:
[tex]\[ y - 7 = 3(x - 1) \\ y - 7 = 3x - 3 \][/tex]
2. Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 3 + 7 \\ y = 3x + 4 \][/tex]
The slope ([tex]\( m \)[/tex]) is 3 and the y-intercept is 4. This does not match our original line equation [tex]\( f(x) = 4x + 3 \)[/tex]. So, the first equation does not represent the same line.
### 2. Equation: [tex]\( y - 1 = 3(x - 7) \)[/tex]
Next, we transform this equation to slope-intercept form:
1. Distribute the 3 on the right-hand side:
[tex]\[ y - 1 = 3(x - 7) \\ y - 1 = 3x - 21 \][/tex]
2. Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 21 + 1 \\ y = 3x - 20 \][/tex]
The slope ([tex]\( m \)[/tex]) here is 3 and the y-intercept is -20. This also does not match our original line equation. Thus, the second equation does not represent the same line.
### 3. Equation: [tex]\( y - 7 = 4(x - 1) \)[/tex]
Let’s rewrite this equation in slope-intercept form:
1. Distribute the 4 on the right-hand side:
[tex]\[ y - 7 = 4(x - 1) \\ y - 7 = 4x - 4 \][/tex]
2. Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 4 + 7 \\ y = 4x + 3 \][/tex]
The slope ([tex]\( m \)[/tex]) here is 4 and the y-intercept is 3. This matches our original line equation [tex]\( f(x) = 4x + 3 \)[/tex]. Therefore, the third equation represents the same line.
### 4. Equation: [tex]\( y - 1 = 4(x - 7) \)[/tex]
Finally, transform this to slope-intercept form:
1. Distribute the 4 on the right-hand side:
[tex]\[ y - 1 = 4(x - 7) \\ y - 1 = 4x - 28 \][/tex]
2. Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 28 + 1 \\ y = 4x - 27 \][/tex]
The slope ([tex]\( m \)[/tex]) is 4 but the y-intercept is -27. This does not match our original line's intercept. Thus, the fourth equation does not represent the same line.
### Conclusion
Out of the four given equations, only [tex]\( y - 7 = 4(x - 1) \)[/tex] represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
