Q. \(7x + 2 = 9\).

Given equation: \(7x + 2 = 9\)

1. Goal: Solve for the variable \(x\). We will use inverse operations to isolate \(x\). Inverse operations undo each other and keep the equality true when applied to both sides.
2. Step 1 — Remove the constant term from the left side by using the additive inverse:There is a constant \(+2\) added to \(7x\). The additive inverse of \(+2\) is \(-2\). Apply the same operation (subtract \(2\)) to both sides of the equation to maintain equality:

   \[7x + 2 – 2 = 9 – 2\]

   On the left side \(+2\) and \(-2\) cancel each other because they are additive inverses, and on the right side we compute the subtraction. Simplify both sides:

   \[7x = 7\]

   Justification: Subtracting the same number from both sides preserves the equality (Property of Equality) and the cancellation arises from \(a + b – b = a\).

3. Step 2 — Isolate \(x\) by removing the coefficient 7 using the multiplicative inverse:The variable \(x\) is multiplied by \(7\). The multiplicative inverse (reciprocal) of \(7\) is \(\tfrac{1}{7}\). Multiply both sides of the equation by \(\tfrac{1}{7}\) or equivalently divide both sides by \(7\):

   \[\frac{1}{7}\cdot(7x) = \frac{1}{7}\cdot 7\]

   On the left side multiplication by the reciprocal cancels the factor \(7\): \(\frac{1}{7}\cdot 7x = x\). On the right side simplify \(\frac{1}{7}\cdot 7 = 1\). So we obtain:

   \[x = 1\]

   Justification: Multiplying both sides of an equation by the same nonzero number preserves equality (Property of Equality). Division by a nonzero number is valid because \(7 \neq 0\).

4. Step 3 — Check the solution by substitution:Substitute \(x = 1\) into the original equation to verify:

   \[7(1) + 2 = 7 + 2 = 9\]

   The left-hand side equals the right-hand side, so the solution satisfies the original equation.

Final answer: \(x = 1\)