To determine which of the given equations represents a line with a [tex]\(y\)[/tex]-intercept of 7, we need to analyze each of the equations individually.
### Given equations:
1. [tex]\(3x - \frac{y}{7} = 0\)[/tex]
2. [tex]\(3x - y - 7 = 0\)[/tex]
3. [tex]\(3x + y - 7 = 0\)[/tex]
#### Step-by-Step Analysis:
1. First Equation: [tex]\(3x - \frac{y}{7} = 0\)[/tex]
Let's rearrange this equation to solve for [tex]\(y\)[/tex]:
[tex]\[
3x - \frac{y}{7} = 0 \implies y = 21x
\][/tex]
In this form [tex]\(y = 21x\)[/tex], there is no constant term. This means the [tex]\(y\)[/tex]-intercept is 0, which does not match our requirement of a [tex]\(y\)[/tex]-intercept of 7. So, this equation cannot be correct.
2. Second Equation: [tex]\(3x - y - 7 = 0\)[/tex]
Let's rearrange this equation to solve for [tex]\(y\)[/tex]:
[tex]\[
3x - y - 7 = 0 \implies y = 3x - 7
\][/tex]
In this slope-intercept form [tex]\(y = mx + c\)[/tex], the term [tex]\(c\)[/tex] gives us the [tex]\(y\)[/tex]-intercept. Here, the [tex]\(y\)[/tex]-intercept is -7, which does not match our requirement of 7. So, this equation cannot be correct either.
3. Third Equation: [tex]\(3x + y - 7 = 0\)[/tex]
Let's rearrange this equation to solve for [tex]\(y\)[/tex]:
[tex]\[
3x + y - 7 = 0 \implies y = -3x + 7
\][/tex]
In this slope-intercept form [tex]\(y = mx + c\)[/tex], the term [tex]\(c\)[/tex] gives us the [tex]\(y\)[/tex]-intercept. Here, the [tex]\(y\)[/tex]-intercept is 7, which exactly matches our requirement.
### Conclusion:
The equation that correctly represents a line with a [tex]\(y\)[/tex]-intercept of 7 is:
[tex]\[ 3x + y - 7 = 0 \][/tex]
So, the correct equation fitting the given [tex]\(y\)[/tex]-intercept is the third equation.
