Solving Two-Step Linear Equations: ax + b = c

A two-step linear equation is an equation that takes exactly two operations to solve for the variable. The general form looks like ax + b = c, where a, b, and c are numbers and x is the variable you're solving for. Unlike a one-step equation, which only needs a single inverse operation, an equation like ax + b = c mixes addition or subtraction with multiplication or division, so you need to undo them in the right order.

Think of ax + b = c as being "built" in a specific order: first x is multiplied by a, then b is added. To solve it, you unbuild it in the opposite order — first undo the addition or subtraction, then undo the multiplication or division. This mirrors reverse PEMDAS, and it works because every step you take must keep both sides of the equation equal.

To solve any equation in the form ax + b = c:

1. Isolate the ax term. Add or subtract b on both sides so that only ax remains on one side.
2. Isolate x. Divide (or multiply) both sides by a to get x by itself.
3. Check. Substitute your answer back into the original equation to confirm both sides match.

Solve 3x + 5 = 20.

Step 1 — subtract 5 from both sides to remove the constant term:

3x + 5 − 5 = 20 − 5, which simplifies to 3x = 15.

Step 2 — divide both sides by 3, the coefficient of x:

3x ÷ 3 = 15 ÷ 3, so x = 5.

Check: 3(5) + 5 = 15 + 5 = 20. It matches the original equation, so x = 5 is correct.

Solve 2x − 7 = 9.

Step 1 — add 7 to both sides, since the constant is being subtracted:

2x − 7 + 7 = 9 + 7, which simplifies to 2x = 16.

Step 2 — divide both sides by 2:

x = 8.

Check: 2(8) − 7 = 16 − 7 = 9. This confirms x = 8 is the solution.

Solve −4x + 3 = −9.

Step 1 — subtract 3 from both sides: −4x = −12.

Step 2 — divide both sides by −4: x = 3.

Check: −4(3) + 3 = −12 + 3 = −9, which matches, so x = 3 is correct. Dividing by a negative number is a common place to slip up, so it's worth writing out this last step carefully.

An equation like 3x + 5 = 20 can be pictured as finding where the line y = 3x + 5 crosses a specific output value. Below, the line y = 3x + 5 is graphed, and the solution x = 5 is the point where the line reaches y = 20.

Sometimes the variable is divided by a number instead, giving a form like x/a + b = c. The idea is the same — undo addition or subtraction first — but the second step becomes multiplication instead of division. That variation is covered in detail in solving two-step linear equations using multiplication and division.

• Dividing before combining constants: always clear the added or subtracted term before dividing by the coefficient.
• Forgetting to apply an operation to both sides: whatever you subtract or divide on one side must happen on the other side too.
• Sign errors with negative coefficients: dividing by a negative number flips the sign of the result, so double-check the arithmetic.
• Skipping the check: substituting your answer back in takes seconds and catches most errors.

Once two-step equations feel comfortable, the next challenge is equations that include parentheses, which are solved using two-step equations with the distributive property. From there, equations with the variable on both sides build directly on the same balancing idea covered here.

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