Solving Two-Step Equations of the Form px + q = r and p(x + q) = r - Seventh Grade Mathematics

Two-step equations combine multiple operations binding the variable, requiring a structured sequence of inverse operations executed in reverse order of operations to achieve isolation.

The Architecture of Two-Step Equations

A two-step equation requires two distinct algebraic inverse operations to isolate the unknown variable. In seventh grade, two canonical algebraic forms predominate:

               THE TWO CANONICAL FORMS
   Form 1:  p * x + q = r        (Multiplication combined with Addition)
   Form 2:  p * (x + q) = r      (Factored grouping scaled by a factor)

In both forms, p, q, and r represent known rational numbers, with p not equal to zero, and x represents the unknown variable.

Solving Form 1: px + q = r (The Shoes and Socks Principle)

When evaluating an arithmetic expression, the order of operations dictates that you multiply before adding. When un-doing operations to solve an equation, you must reverse the order: undo addition or subtraction first, then undo multiplication or division.

               THE SHOES AND SOCKS ANALOGY
   Putting them on (Evaluation):   Socks first (multiply), then Shoes (add)
   Taking them off (Solving):      Shoes first (subtract), then Socks (divide)

To solve px + q = r: Step 1: Eliminate the constant term q by adding or subtracting its opposite on both sides. Step 2: Eliminate the coefficient p by dividing both sides by p (or multiplying by its reciprocal).

Example: Solve -3x + 14 = -7
Step 1 (Undo addition of 14):
         -3x + 14 - 14 = -7 - 14
         -3x = -21
Step 2 (Undo multiplication by -3):
         -3x / (-3) = -21 / (-3)
         x = 7

Check: -3(7) + 14 = -21 + 14 = -7 (Verified).

Solving Form 2: p(x + q) = r

Equations of the form p(x + q) = r feature a parenthetical grouping scaled by an outside factor p. There are two mathematically valid strategies to solve this structure:

METHOD A: Divide First (Most efficient when r is divisible by p)
  p * (x + q) = r
  Divide both sides by p:     x + q = r / p
  Subtract q from both sides: x = (r / p) - q

METHOD B: Distribute First (Always reliable, especially with fractions)
  p * (x + q) = r
  Distribute p:               px + pq = r
  Subtract pq:                px = r - pq
  Divide by p:                x = (r - pq) / p

Let us contrast both methods solving two times the quantity x minus five equals twenty-six:

Method A (Divide first):
  2 * (x - 5) = 26
  (x - 5) = 26 / 2
  x - 5 = 13
  x = 13 + 5 = 18

Method B (Distribute first):
  2x - 10 = 26
  2x = 26 + 10
  2x = 36
  x = 36 / 2 = 18

Both methods arrive at the identical solution, x equals eighteen.

Handling Equations with Rational Numbers

When equations feature fractional coefficients or constants, clearing denominators or multiplying by reciprocals maintains computational clarity.

Example: Solve (2/5)x - 3 = 7
Step 1 (Add 3):         (2/5)x = 7 + 3 = 10
Step 2 (Reciprocal):    x = 10 * (5/2) = (10 / 2) * 5 = 5 * 5 = 25

Multiplying by the reciprocal five-halves isolates x cleanly in a single step.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Solve the two-step equation negative four x plus nine equals negative nineteen.

Exercise 2. Solve the equation with a fractional coefficient: one-third x minus eight equals negative two.

Exercise 3. Solve the factored equation negative five times the quantity x plus six equals forty-five using the divide-first method.

Exercise 4. Solve the equation zero point four y plus two point eight equals six point four.

Solutions

Solution to Exercise 1. The equation is negative four x plus nine equals negative nineteen. First undo the addition of nine by subtracting nine from both sides: negative nineteen minus nine equals negative twenty-eight, yielding negative four x equals negative twenty-eight. Next undo the multiplication by negative four by dividing both sides by negative four: negative twenty-eight divided by negative four equals positive seven. The solution is x equals seven. Check: negative four times seven is negative twenty-eight; negative twenty-eight plus nine is negative nineteen.

Solution to Exercise 2. The equation is one-third x minus eight equals negative two. First undo the subtraction of eight by adding eight to both sides: negative two plus eight equals positive six, yielding one-third x equals six. Next undo multiplication by one-third by multiplying both sides by three (the reciprocal): six times three equals eighteen. The solution is x equals eighteen. Check: one-third of eighteen is six; six minus eight is negative two.

Solution to Exercise 3. The equation is negative five times the quantity x plus six equals forty-five. Using the divide-first method, divide both sides of the equation by negative five: forty-five divided by negative five equals negative nine, yielding x plus six equals negative nine. Now isolate x by subtracting six from both sides: negative nine minus six is equivalent to negative nine plus negative six, which equals negative fifteen. The solution is x equals negative fifteen. Check: negative fifteen plus six is negative nine; negative five times negative nine equals positive forty-five.

Solution to Exercise 4. The equation is zero point four y plus two point eight equals six point four. First subtract two point eight from both sides: six point four minus two point eight equals three point six, yielding zero point four y equals three point six. Next divide both sides by zero point four: three point six divided by zero point four is equivalent to thirty-six divided by four, which equals nine. The solution is y equals nine. Check: zero point four times nine is three point six; three point six plus two point eight equals six point four.