Of course! Let's tackle each part of the question step-by-step.
### Part (i)
We are given the formula:
[tex]\[ W = -\frac{3}{2} A - 4B + C \][/tex]
and we need to make \( B \) the subject of the formula.
1. Start with the given equation:
[tex]\[ W = -\frac{3}{2} A - 4B + C \][/tex]
2. Subtract \( C \) from both sides to isolate the term involving \(B\):
[tex]\[ W - C = -\frac{3}{2} A - 4B \][/tex]
3. Next, add \(\frac{3}{2} A\) to both sides to further isolate \(B\):
[tex]\[ W - C + \frac{3}{2} A = -4B \][/tex]
4. Now, divide both sides by \(-4\) to solve for \( B \):
[tex]\[ B = -\frac{1}{4}(W - C + \frac{3}{2}A) \][/tex]
Simplifying the expression inside the parentheses:
[tex]\[ B = -\frac{1}{4}W + \frac{1}{4}C - \frac{3}{8}A \][/tex]
Thus, \( B \) in terms of \( A \), \( C \), and \( W \) is:
[tex]\[ B = -0.25 W + 0.25 C - 0.375 A \][/tex]
### Part (ii)
We are given the formula:
[tex]\[ a = -b + \sqrt{m - 1} - c \][/tex]
and we need to make \( m \) the subject of the formula.
1. Start with the given equation:
[tex]\[ a = -b + \sqrt{m - 1} - c \][/tex]
2. Add \( b \) to both sides to move \(-b\) across the equation:
[tex]\[ a + b = \sqrt{m - 1} - c \][/tex]
3. Next, add \( c \) to both sides to isolate the square root term:
[tex]\[ a + b + c = \sqrt{m - 1} \][/tex]
4. Square both sides to eliminate the square root and solve for \( m \):
[tex]\[ (a + b + c)^2 = m - 1 \][/tex]
5. Finally, add \( 1 \) to both sides to solve for \( m \):
[tex]\[ m = (a + b + c)^2 + 1 \][/tex]
So, \( m \) in terms of \( a \), \( b \), and \( c \) is:
[tex]\[ m = (a + b + c)^2 + 1 \][/tex]
Thus, our final answers are:
(i) \( B = -0.25 W + 0.25 C - 0.375 A \)
(ii) [tex]\( m = (a + b + c)^2 + 1 \)[/tex]
