2 Change the subject of the formula to \( x \) 7.2.1 \( m=\sqrt{x^{2}}+a x \) 7.2.2 \( Q=m c(y-x) \) 7.2.3 \( m(m x-n)-n(n x-m)=m^{2} n^{2} \)

Below are the steps to “solve for x” (i.e. change the subject of the formula to x) for each formula. Note that in the first formula the square root of x² is |x|, so there are two cases unless you assume x is nonnegative. ────────────────────────────── 7.2.1 m = √(x²) + a·x Since √(x²) = |x|, you have two cases: • Case 1. Assume x ≥ 0 so that |x| = x:   m = x + a·x = (1 + a)x.   Assuming 1 + a ≠ 0, solve for x:    x = m/(1 + a). • Case 2. Assume x < 0 so that |x| = –x:   m = (–x) + a·x = (a – 1)x.   Assuming a – 1 ≠ 0, solve for x:    x = m/(a – 1). If additional context indicates x is nonnegative, then x = m/(1 + a) is the answer. ────────────────────────────── 7.2.2 Q = m·c·(y – x) Solve for x by isolating the (y – x) term:   Divide both sides by m·c (assuming m·c ≠ 0):    y – x = Q/(m·c)   Subtract Q/(m·c) from y:    x = y – Q/(m·c) ────────────────────────────── 7.2.3 m(m·x – n) – n(n·x – m) = m²n² Step 1. Expand the left‐hand side:   m(m·x – n) = m²x – m·n,   n(n·x – m) = n²x – n·m. So the equation becomes:   (m²x – m·n) – (n²x – n·m) = m²n² Step 2. Distribute the minus sign:   m²x – m·n – n²x + n·m = m²n² Notice that –m·n + n·m = 0, so the x–terms combine:   (m² – n²)x = m²n² Step 3. Solve for x (provided m² – n² ≠ 0):   x = (m²n²) / (m² – n²) ────────────────────────────── Summary of Answers: 7.2.1 • For x ≥ 0: x = m/(1 + a)      • For x < 0: x = m/(a – 1) 7.2.2  x = y – Q/(m·c) 7.2.3  x = m²n² / (m² – n²) These are the required forms with x as the subject of each formula.