Solving Multistep Linear Equations

Objective: Identify a linear equation and use equivalent operations to isolate its variable.

Prerequisites: signed-number arithmetic, order of operations, combining like terms, and the distributive property. A linear equation in x can be simplified to ax + b = c, with a nonzero. A solution makes both sides equal. Adding or subtracting the same quantity on both sides, or multiplying or dividing both sides by the same nonzero constant, preserves the solution set. Simplify each side first, undo addition or subtraction, and then undo multiplication or division. Do not move a term by a memorized sign change without understanding the operation on both sides.

Objective: Distribute accurately and combine like terms to solve a multistep linear equation.

Use a(bx + c) = abx + ac: the outside factor multiplies every term inside parentheses. A negative factor changes the sign of each product. Combine terms only when they have the same variable part. Constants combine with constants. When variables occur on both sides, subtract the variable term with the smaller coefficient when convenient, then isolate the remaining term. Every intermediate equality should describe the same solution. Dividing an equation with several terms requires dividing every term, not just the variable term. Classification of identities and contradictions is developed in 2.4; here every practice equation has a nonzero resulting coefficient.

Objective: Solve more involved linear equations, report the requested expression, and verify algebra or a graphing result.

Read the target before calculating: a question may ask for 2x + 1 rather than x. After solving, substitute into that target and into the original equation. In contexts, attach units and check that the result is feasible. Algebra usually gives an exact answer quickly for a short linear equation. For a graphing check in Desmos, enter y equal to the left expression and y equal to the right expression, then inspect their intersection; its x-coordinate is the solution, while its y-coordinate is the shared value. Adjust the window if the intersection is outside view. A decimal graph display can be rounded: verify exact fractions algebraically. A horizontal line is valid for a constant side. No graph or diagram is needed for this lesson.