Let's analyze and convert the given line equation to the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the [tex]\(y\)[/tex]-intercept.
The given equation is:
[tex]\[
2y + 5x = -7
\][/tex]
First, isolate [tex]\(y\)[/tex] on one side of the equation. Begin by moving the [tex]\(5x\)[/tex] term to the right side:
[tex]\[
2y = -5x - 7
\][/tex]
Next, divide every term by 2 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \left(-\frac{5}{2}\right)x - \frac{7}{2}
\][/tex]
Now the equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope [tex]\(m\)[/tex] is the coefficient of [tex]\(x\)[/tex], and the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is the constant term.
From this equation, we can see:
- The slope [tex]\(m\)[/tex] is [tex]\(-\frac{5}{2}\)[/tex].
- The [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is [tex]\(-\frac{7}{2}\)[/tex].
Therefore, in coordinate form, the [tex]\(y\)[/tex]-intercept is [tex]\((0, -\frac{7}{2})\)[/tex].
Given the options:
(A) [tex]\(m = 5\)[/tex], [tex]\(y\)[/tex]-intercept [tex]\((0, -7)\)[/tex]
(B) [tex]\(m = -\frac{5}{2}\)[/tex], [tex]\(y$-intercept $\left(0, -\frac{7}{2}\right)\)[/tex]
(C) [tex]\(m = -7\)[/tex], [tex]\(y\)[/tex]-intercept [tex]\((0, 5)\)[/tex]
(D) [tex]\(m = -\frac{7}{2}\)[/tex], [tex]\(y\)[/tex]-intercept [tex]\(\left(0, -\frac{5}{2}\right)\)[/tex]
The correct choice based on the calculated slope and [tex]\(y\)[/tex]-intercept is:
(B) [tex]\(m = -\frac{5}{2}\)[/tex], [tex]\(y$-intercept $\left(0, -\frac{7}{2}\right)\)[/tex]
Additionally, you can match these values with the given results:
- The slope is [tex]\(-2.5\)[/tex] (which is equivalent to [tex]\(-\frac{5}{2}\)[/tex]).
- The [tex]\(y$-intercept is \(-3.5\)[/tex] (which is equivalent to [tex]\(-\frac{7}{2}\)[/tex]).
Therefore, the correct answer based on the calculations is (B).
