Eighth Grade Mathematics: Multi-Step Equations with Rational Coefficients

Solving an algebraic equation is fundamentally an exercise in maintaining balance: whatever operation is performed on one side of the equal sign must be performed identically on the other side. As equations become more intricate, incorporating parentheses, variable terms on a single side, and rational coefficients like fractions and decimals, you must develop a structured roadmap to isolate the unknown variable. In this chapter, you will master the systematic multi-step equation-solving algorithm and apply it to equations featuring rational numbers.

The Multi-Step Equation-Solving Roadmap

Every linear equation in one variable can be solved by executing a sequence of four standardized algebraic phases:

Phase 1: Simplify Each Side Independently
         - Distribute to eliminate all parentheses: a*(b + c) = ab + ac
         - Combine like terms on the left side; combine like terms on the right side.

Phase 2: Isolate Variable Terms
         - Use addition or subtraction properties of equality to collect all variable
           terms onto one side of the equation.

Phase 3: Isolate Constant Terms
         - Use addition or subtraction properties of equality to collect all constant
           numerical terms on the opposite side of the equation.

Phase 4: Solve for the Variable
         - Use multiplication or division properties of equality to make the variable's
           coefficient equal to 1.

At every step of this process, the equation remains balanced because the properties of equality guarantee that applying identical operations preserves numerical truth.

Working with Fractional and Decimal Coefficients

Equations involving fractions and decimals follow the exact same logic as equations with integers, but require precision in fraction arithmetic and decimal place values.

Solve: (2/3)*(6x - 9) + 4 = 18

Step 1: Distribute (2/3) into (6x - 9):
        (2/3)*(6x) - (2/3)*(9) + 4 = 18
        (12/3)x - (18/3) + 4 = 18
        4x - 6 + 4 = 18

Step 2: Combine like constants on the left side:
        4x - 2 = 18

Step 3: Add 2 to both sides to isolate the variable term:
        4x = 20

Step 4: Divide both sides by 4:
        x = 5

Solve: 0.4x + 1.2 = 3.6 - 0.2x

Step 1: Sides are already simplified. Add 0.2x to both sides:
        0.6x + 1.2 = 3.6

Step 2: Subtract 1.2 from both sides:
        0.6x = 2.4

Step 3: Divide both sides by 0.6:
        x = 2.4 / 0.6 = 4

Notice that dividing 2.4 by 0.6 is equivalent to dividing 24 by 6, yielding exactly 4.

Checking Solutions via Direct Substitution

A solution to an equation is a value that makes the equation true. You should never guess whether your answer is correct; you can verify it with mathematical certainty by substituting the candidate value back into the original equation and evaluating both sides.

Check x = 5 in: (2/3)*(6x - 9) + 4 = 18
Left Side:  (2/3)*(6*(5) - 9) + 4
          = (2/3)*(30 - 9) + 4
          = (2/3)*(21) + 4
          = 14 + 4 = 18
Right Side: 18

Both sides evaluate to 18! The solution x = 5 is mathematically verified.

If the left side and right side do not evaluate to identical values, either an arithmetic error occurred during the solution steps or during the check itself.

Chapter Exercises

Exercise 1: Solve the multi-step linear equation 3*(2x - 5) - 4x + 7 = 16. Show each step of distribution, combining like terms, isolating the variable, and checking your solution.

Exercise 2: Solve the equation (3/4)*(8x - 12) + 5 = 26. Show all fraction multiplications clearly and state the final value of x.

Exercise 3: Solve the decimal equation 0.15*(2x + 20) - 0.25x = 2.1. Explain each step in paragraph form and verify the final solution.

Solutions and Explanations

Solution 1: To solve 3(2x - 5) - 4x + 7 = 16, start by simplifying the left side of the equation. Apply the distributive property to 3(2x - 5): multiplying 3 by 2x yields 6x, and multiplying 3 by -5 yields -15, resulting in 6x - 15 - 4x + 7 = 16. Next, combine like terms on the left side: combine the variable terms 6x - 4x to get 2x, and combine the constants -15 + 7 to get -8. This simplifies the equation to 2x - 8 = 16. Now isolate the variable term by adding 8 to both sides: 2x = 16 + 8, which gives 2x = 24. Finally, divide both sides by 2 to isolate x: x = 24 / 2 = 12. To check, substitute x = 12 into the original equation: 3(2(12) - 5) - 4(12) + 7 = 3(24 - 5) - 48 + 7 = 3*(19) - 48 + 7 = 57 - 48 + 7 = 9 + 7 = 16. Since the left side equals the right side (16 = 16), the solution x = 12 is verified.

Solution 2: To solve (3/4)(8x - 12) + 5 = 26, begin by distributing the fraction 3/4 across the terms in parentheses. Multiply (3/4) by 8x: (3 * 8 / 4)x = (24/4)x = 6x. Multiply (3/4) by -12: -(3 * 12 / 4) = -(36/4) = -9. Rewriting the equation gives 6x - 9 + 5 = 26. Next, combine the constant terms on the left side: -9 + 5 = -4, giving 6x - 4 = 26. Add 4 to both sides of the equation to isolate the variable term: 6x = 30. Finally, divide both sides by 6: x = 30 / 6 = 5. Checking the solution: (3/4)(8(5) - 12) + 5 = (3/4)(40 - 12) + 5 = (3/4)*(28) + 5 = 21 + 5 = 26, which matches the right side perfectly.

Solution 3: To solve 0.15(2x + 20) - 0.25x = 2.1, start by distributing 0.15 into the parentheses: 0.15 multiplied by 2x gives 0.30x (or 0.3x), and 0.15 multiplied by 20 gives 3.0. The equation becomes 0.30x + 3.0 - 0.25x = 2.1. Next, combine the like variable terms on the left side: 0.30x - 0.25x equals 0.05x. This yields 0.05x + 3.0 = 2.1. Now subtract the constant 3.0 from both sides: 0.05x = 2.1 - 3.0, which yields 0.05x = -0.9. Finally, divide both sides by the coefficient 0.05: x = -0.9 / 0.05. Multiplying numerator and denominator by 100 simplifies this to -90 / 5, which equals -18. To verify the solution: substitute x = -18 into the original equation: 0.15(2(-18) + 20) - 0.25(-18) = 0.15(-36 + 20) - (-4.5) = 0.15(-16) + 4.5 = -2.4 + 4.5 = 2.1. The solution x = -18 is confirmed.