Sure, let's go through the problem step-by-step to compare the given equation [tex]\(x - \frac{y}{2} + 8 = 0\)[/tex] with the standard form [tex]\(ax + by + c = 0\)[/tex], and then find the value of [tex]\(2a + b - c\)[/tex].
1. Identify the coefficients:
The given equation is [tex]\(x - \frac{y}{2} + 8 = 0\)[/tex].
To identify coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], compare the given equation to the standard form [tex]\(ax + by + c = 0\)[/tex]:
- The coefficient of [tex]\(x\)[/tex] is [tex]\(a = 1\)[/tex].
- The coefficient of [tex]\(y\)[/tex] is [tex]\(b = -\frac{1}{2}\)[/tex].
- The constant term [tex]\(c = 8\)[/tex].
2. Calculate [tex]\(2a + b - c\)[/tex]:
Now that we have the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we can substitute them into the expression [tex]\(2a + b - c\)[/tex].
Substitute [tex]\(a = 1\)[/tex], [tex]\(b = -\frac{1}{2}\)[/tex], and [tex]\(c = 8\)[/tex] into the expression:
[tex]\[
2a + b - c = 2(1) + \left(-\frac{1}{2}\right) - 8
\][/tex]
3. Simplify the expression:
Step-by-step calculation:
[tex]\[
= 2 \cdot 1 + \left(-\frac{1}{2}\right) - 8
\][/tex]
First, calculate [tex]\(2 \cdot 1 = 2\)[/tex]:
[tex]\[
= 2 + \left(-\frac{1}{2}\right) - 8
\][/tex]
Then, add [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[
= 2 - \frac{1}{2} - 8
\][/tex]
Convert 2 to a fraction for easier subtraction:
[tex]\[
= \frac{4}{2} - \frac{1}{2} - 8
\][/tex]
Subtract the fractions:
[tex]\[
= \frac{4}{2} - \frac{1}{2} = \frac{3}{2}
\][/tex]
Finally, subtract 8:
[tex]\[
\frac{3}{2} - 8 = \frac{3}{2} - \frac{16}{2} = \frac{3 - 16}{2} = \frac{-13}{2} = -6.5
\][/tex]
So, the value of [tex]\(2a + b - c\)[/tex] is [tex]\(-6.5\)[/tex].
