Two-Step Equations

⚖️ Two-Step Equations

Part 1 of 5 — What "Undoing" Means

Topics in This Part

🔑 Key Concept: Solving an equation means getting the variable all by itself on one side. A two-step equation just means it takes two undo-moves to get there. Part 1 builds the one habit the whole lesson depends on: every operation has an opposite, and we undo them in reverse order.

What Is a Two-Step Equation?

A one-step equation is solved with a single move:

$x + 5 = 12 \quad\Longrightarrow\quad x = 7 \quad(\text{subtract } 5)$

A two-step equation has the variable tangled up in two operations, so it takes two moves to free it. Most look like this:

$2x + 3 = 11$

Here $x$ is (1) multiplied by 2 and then (2) added to 3. To solve it, we have to undo both — that is the "two steps."

💡 Spotting them: If you see a number stuck next to the variable (like the $2$ in $2x$) and a number being added or subtracted (like the $+3$), it's a two-step equation.

Inverse Operations: Every Move Has an "Undo"

To get rid of an operation, you do its inverse (its opposite) to both sides of the equation. Keeping both sides equal is the golden rule — an equation is a balanced scale ⚖️.

Example (one step): $x - 8 = 5$. The $8$ is subtracted, so we add $8$ to both sides:

$x - 8 + 8 = 5 + 8 \quad\Longrightarrow\quad x = 13$

⚠️ Both sides, every time. Whatever you do to one side, you must do to the other — otherwise the scale tips and the equation is no longer true.

Concept Check 🎯

The Order of Operations, Run Backwards

Remember PEMDAS — the order you build an expression: multiply/divide before you add/subtract. To take it apart, you go in the reverse order.

Think of getting dressed: socks then shoes. To undo it, you take off shoes first, then socks. Same idea here.

So for a two-step equation, the rule is:

🔑 The Golden Order: Undo addition and subtraction first, then undo multiplication and division. (Reverse of PEMDAS.)

Why? In $2x + 3 = 11$, the expression was built by multiplying ($2x$) and then adding ($+3$). Undoing in reverse means we strip off the $+3$ first, then the $\times 2$.

Plan the Moves 🔽

For each equation, choose the operation you would undo first.

Warm-Up: One Move at a Time

Before we chain two moves together in Part 2, get comfortable with a single inverse. Each of these is just one undo away from solved.

• $x - 6 = 10$ → add $6$ → $x = 16$
• $\dfrac{x}{2} = 9$ → multiply by $2$ → $x = 18$
• $4x = 20$ → divide by $4$ → $x = 5$

💡 A two-step equation is really just two of these in a row. Master one move, and two becomes easy.

One-Step Warm-Up 🧮

Solve each with a single inverse operation. Enter the value of $x$.

1) $x - 7 = 5,\quad x = \,?$ 2) $3x = 18,\quad x = \,?$ 3) $\dfrac{x}{6} = 4,\quad x = \,?$

⚖️ Two-Step Equations

Part 2 of 5 — The Core Method: $px + q = r$

🔑 The Plan, every time: Step 1. Undo the addition or subtraction (the $+q$ or $-q$). Step 2. Undo the multiplication or division stuck to the variable. Two moves, same order, every problem.

Worked Example: $2x + 3 = 11$

Step 1 — undo the $+3$. Subtract $3$ from both sides:

$2x + 3 - 3 = 11 - 3 \quad\Longrightarrow\quad 2x = 8$

Step 2 — undo the $\times 2$. Divide both sides by $2$:

$\frac{2x}{2} = \frac{8}{2} \quad\Longrightarrow\quad x = 4$

✅ Check: Put $x = 4$ back in: $2(4) + 3 = 8 + 3 = 11$ ✓ — it matches, so $x = 4$ is correct.

Worked Example: $5x - 7 = 18$

Step 1 — undo the $-7$. Add $7$ to both sides:

$5x - 7 + 7 = 18 + 7 \quad\Longrightarrow\quad 5x = 25$

Step 2 — undo the $\times 5$. Divide both sides by $5$:

$\frac{5x}{5} = \frac{25}{5} \quad\Longrightarrow\quad x = 5$

✅ Check: $5(5) - 7 = 25 - 7 = 18$ ✓

💡 Pattern: Notice both problems followed the exact same two moves. Once the order is a habit, two-step equations become routine.

Concept Check 🎯

Your Turn — Same Two Moves

You've seen the pattern twice. Now drive it yourself on the next set. For every problem, say it out loud:

"Step 1: undo the plus-or-minus. Step 2: divide by the number in front of $x$."

Keep your work in two short lines, and don't forget you can always check by substituting your answer back in.

Solve It 🧮

Solve each two-step equation for $x$. Use the two moves in order.

1) $2x + 9 = 21,\quad x = \,?$ 2) $6x - 4 = 26,\quad x = \,?$ 3) $3x + 7 = 7,\quad x = \,?$

See Every Step Laid Out

It helps to watch the two moves happen one line at a time. Here is $4x + 6 = 30$ with nothing skipped:

$4x + 6 = 30$ $4x + 6 - 6 = 30 - 6 \quad(\text{Step 1: subtract } 6)$ $4x = 24$ $\frac{4x}{4} = \frac{24}{4} \quad(\text{Step 2: divide by } 4)$ $x = 6$

💡 Writing the subtraction and division on both sides keeps the scale balanced and makes mistakes easy to spot. In the next check, you'll choose each step yourself.

Build the Solution 🔽

Solve $4x + 6 = 30$ by choosing what belongs at each stage.

⚖️ Two-Step Equations

Part 3 of 5 — Negatives, Division & Checking

🔑 Same method, trickier numbers. Real Grade 7 problems involve negative numbers, a variable being divided (like $\frac{x}{3}$), and sometimes answers that are negative or fractions. The two moves never change — we just have to handle signs carefully.

When Negatives Show Up

Signs trip up more students than anything else, so go slow and keep the sign attached to its number.

Worked Example: $-3x + 5 = 14$

Step 1 — undo the $+5$. Subtract $5$:

$-3x = 14 - 5 = 9$

Step 2 — undo the $\times(-3)$. Divide by $-3$:

$x = \frac{9}{-3} = -3$

✅ Check: $-3(-3) + 5 = 9 + 5 = 14$ ✓ — remember, a negative times a negative is positive.

⚠️ Watch the sign! Dividing by a negative coefficient flips the sign of your answer. $9 \div (-3) = -3$, not $+3$.

Concept Check 🎯

When the Variable Is Divided

Sometimes the variable is divided by a number, like $\dfrac{x}{4}$. Its inverse is multiplication, so Step 2 changes from "divide" to "multiply."

Worked Example: $\dfrac{x}{4} - 2 = 3$

Step 1 — undo the $-2$. Add $2$:

$\frac{x}{4} = 3 + 2 = 5$

Step 2 — undo the $\div 4$. Multiply both sides by $4$:

$x = 5 \cdot 4 = 20$

✅ Check: $\dfrac{20}{4} - 2 = 5 - 2 = 3$ ✓

💡 Key swap: If the variable is divided, you multiply in Step 2. If it is multiplied, you divide. The plan is the same; only the inverse changes.

Solve It 🧮

Solve for $x$. Watch for division and negatives. Fractions like $-5/12$ and decimals are fine.

1) $\dfrac{x}{5} + 6 = 10,\quad x = \,?$ 2) $\dfrac{x}{2} - 7 = -3,\quad x = \,?$ 3) $-5x - 1 = 14,\quad x = \,?$

Always Check Your Answer

Checking is not optional — it's how you prove you're right and catch sign slips. Substitute your answer back into the original equation and confirm both sides match.

Example: did we solve $\dfrac{x}{3} + 4 = 6$ correctly with $x = 6$?

$\frac{6}{3} + 4 = 2 + 4 = 6 \;?\; \text{...but the right side is } 6. \;✓$

Wait — that checks. But suppose a student got $x = 18$ instead:

$\frac{18}{3} + 4 = 6 + 4 = 10 \ne 6 \quad ✗$

The check catches the mistake instantly. The correct answer is $x = 6$ (subtract 4 → $\frac{x}{3} = 2$ → multiply by 3 → $x = 6$).

🔑 Make checking a habit. Thirty seconds of substitution saves you from a wrong answer you felt sure about.

Check the Work 🔽

A student solved each equation. Use a check to decide if the answer is correct or wrong.

⚖️ Two-Step Equations

Part 4 of 5 — Real-World Word Problems

🔑 The real skill: turning a sentence into an equation. Most two-step word problems follow the pattern "a starting/fixed amount, plus a rate times an unknown." That's exactly $px + q = r$ — a perfect two-step equation.

Translating Words into $px + q = r$

Look for three roles in the story:

The variable $x$ is the unknown count — how many, how long, how much.

Example sentence

"A taxi charges a $4 flat fee plus $2 per mile. The total ride cost $16. How many miles was it?"

• Flat fee $q = 4$, rate $p = 2$, total $r = 16$, miles $= x$.

$2x + 4 = 16$

Concept Check 🎯

Worked Example: The Taxi Problem

"A taxi charges a $4 flat fee plus $2 per mile. The ride cost $16. How many miles?"

Set up: $2x + 4 = 16$

Step 1 — undo the $+4$: $2x = 16 - 4 = 12$

Step 2 — undo the $\times 2$: $x = \frac{12}{2} = 6$

Answer: the ride was 6 miles.

✅ Check in context: $2(6) + 4 = 12 + 4 = 16$ dollars ✓. Always ask: does this answer make sense? Six miles for a $16 taxi ride is reasonable. 👍

💡 Label your answer with units (miles, dollars, months). A naked number "6" is incomplete in a word problem — "6 miles" tells the whole story.

Word-Problem Practice 🧮

Set up a two-step equation and solve. Enter only the number.

1) A pizza costs $3 per topping plus a $8 base price. A pizza cost $20. How many toppings? 2) Jordan saved $50, then added $25 each week. Now she has $200. How many weeks? 3) A printer prints a 2-page cover, then 4 pages per chapter, for 30 pages total. How many chapters?

Watch the Signal Words

The wording tells you which number is the rate and which is the constant:

⚠️ Don't mix them up. A "$3 fee plus $2 per item" becomes $2x + 3$ — the per-item $2 multiplies $x$, while the one-time $3 is added on. Swapping them ($3x + 2$) gives a different, wrong equation.

Match the Story to the Equation 🔽

"A delivery service charges a $6 fee plus $3 per item. The bill was $27 for $n$ items."

⚖️ Two-Step Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) spot a two-step equation, (2) undo it in the right order, (3) handle negatives and division, and (4) build an equation from a word problem. Let's put it all together and finish with an Exit Quiz.

Quick Reference

⚠️ Two classics to avoid: (1) dividing before you subtract — always undo $+/-$ first; (2) losing a negative sign — keep the sign glued to its number, and check by substituting.

Mixed Practice 🧮

Solve for $x$. Mix of multiply, divide, and negatives.

1) $7x - 9 = 33,\quad x = \,?$ 2) $\dfrac{x}{3} + 5 = 1,\quad x = \,?$ 3) $-4x + 10 = 2,\quad x = \,?$

The Three Mistakes That Cost Points

When students miss two-step equations, it's almost always one of these:

1. Wrong order — dividing before subtracting. Undo $+/-$ first.
2. A dropped sign — turning $-4x = -8$ into $x = -2$ instead of $x = 2$. Keep the sign with its number.
3. No check — being "pretty sure" instead of certain. Substitute back, every time.

The next set mixes pure equations with one word problem — exactly what a test looks like.

🔑 If you slow down on the order and the signs, two-step equations become some of the most reliable points you can earn.

Mixed Practice 🎯

You're Ready 🎓

Three skills, all in one place: undo in the right order, mind the signs, and translate a story into $px + q = r$. The Exit Quiz below has one of each — a plain equation, a negative, and a word problem.

💡 Take your time, do both steps, and check each answer before you move on.

Exit Quiz ✅

Answer all three to finish the lesson.