🎯⭐ INTERACTIVE LESSON

Solving Two-Step Equations

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Solving Two-Step Equations - Complete Interactive Lesson

Part 1: The Big Idea: Undoing Operations

⚖️ Solving Two-Step Equations

Part 1 of 5 — The Big Idea: Undoing Operations

Topics in This Part

Section
What a Two-Step Equation Looks Like
Inverse Operations (Undoing)
The Order: Undo in Reverse

🔑 Key Concept: Solving an equation means getting the variable alone. A two-step equation just took two operations to build, so it takes two moves to undo — done in the reverse order.

What Is a Two-Step Equation?

A two-step equation has the variable wrapped in two operations. The classic form is:

$ax + b = c$

Here the variable $x$ is first multiplied by $a$ , then a number $b$ is added. To solve it you must undo both.

One step vs. two steps

Equation	Steps to build it	Type
$x + 7 = 12$	add $7$	one-step
$3x = 15$	multiply by

💡 If you see a number stuck to the variable by multiplication and a separate number added or subtracted, it's a two-step equation.

Concept Check 🎯

Inverse Operations: The Undo Buttons

Every operation has an inverse that cancels it out. To free the variable, replace each operation with its inverse.

Operation	Inverse (undo)
Addition $(+)$	Subtraction $(-)$
Subtraction $(-)$	Addition $(+)$

Pick the Undo 🔽

For each operation, choose the inverse operation that cancels it.

The Order: Undo in REVERSE

When you build an expression, you follow order of operations (multiply before adding). When you undo it, you go backwards.

Think of getting dressed: socks then shoes. To undo, you take off shoes then socks — reverse order!

To solve $ax + b = c$ :

First undo the $+b$ or $-b$ (the addition/subtraction). ➡️

Order Check 🎯

Part 2: The Standard Move: $ax + b = c$

⚖️ Solving Two-Step Equations

Part 2 of 5 — The Standard Move: $ax + b = c$

🔑 The Recipe: (1) Subtract or add to clear the constant, (2) Divide by the coefficient, (3) Check by plugging back in.

The Two-Step Recipe

To solve $ax + b = c$ :

Part 3: Division, Negatives & Variable Positions

⚖️ Solving Two-Step Equations

Part 3 of 5 — Division, Negatives & Variable Positions

🔑 Same recipe, new disguises: equations with $\dfrac{x}{a}$ , a negative coefficient, or the constant written first all use the very same two-step plan.

When the Variable Is Divided

If the equation has $\dfrac{x}{a}$ , the variable is by , so you undo it by by .

Part 4: Fractions, Decimals & Word Problems

⚖️ Solving Two-Step Equations

Part 4 of 5 — Fractions, Decimals & Word Problems

🔑 Real numbers, real problems: the two-step method handles fractional coefficients and translates straight from everyday situations like phone plans and savings goals.

Fractional Coefficients

When the variable is multiplied by a fraction like $\dfrac{2}{3}$ , the fastest undo is to multiply by its reciprocal $\dfrac{3}{2}$ (flip the fraction).

Part 5: Mixed Practice & Mastery Check

⚖️ Solving Two-Step Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize two-step equations, (2) undo in reverse order, (3) handle division, negatives, fractions, and decimals, and (4) translate word problems. Let's put it all together.

Quick Reference

Equation type	First move	Second move
$ax + b = c$	subtract $b$	divide by

🔑 Golden Rule of Equations: Whatever you do to one side, you must do to the other side. That keeps the equation balanced — like a scale. ⚖️

clears the loose number

Then undo the $\times a$ or $\div a$ (the multiplication/division). ➡️ frees the variable

⚠️ Common mistake: dividing before you've removed the added number. Always peel off the addition/subtraction first, then deal with the multiplication/division.

Subtract $b$ from both sides → $ax = c - b$
Divide by $a$ on both sides → $x = \dfrac{c - b}{a}$
Check by substituting your answer back in.

Worked Example: $3x + 7 = 22$

$3x + 7 = 22$ $3x + 7 - 7 = 22 - 7 \quad\text{(subtract 7)}$ $3x = 15$ $\frac{3x}{3} = \frac{15}{3} \quad\text{(divide by 3)}$ $x = 5$

✅ Check: $3(5) + 7 = 15 + 7 = 22$ ✓

Worked Example: $5x - 4 = 26$

This time the constant is subtracted, so we add to undo it:

$5x - 4 = 26$ $5x - 4 + 4 = 26 + 4 \quad\text{(add 4)}$ $5x = 30$ $x = \frac{30}{5} = 6$

✅ Check: $5(6) - 4 = 30 - 4 = 26$ ✓

Worked Example: $2x + 9 = 3$

Answers can be negative — that's perfectly fine:

$2x + 9 = 3 \;\Rightarrow\; 2x = 3 - 9 = -6 \;\Rightarrow\; x = \frac{-6}{2} = -3$

✅ Check: $2(-3) + 9 = -6 + 9 = 3$ ✓

Concept Check 🎯

Order the Steps 🔽

You're solving $7x + 2 = 30$ . Choose what happens at each stage.

Solve It 🧮

Solve each equation for $x$ . Enter just the number.

1) $2x + 6 = 14 \;\Rightarrow\; x = \,?$ 2) $8x - 3 = 29 \;\Rightarrow\; x = \,?$ 3) $5x + 12 = 2 \;\Rightarrow\; x = \,?$

Worked Example: $\dfrac{x}{4} - 2 = 5$

$\frac{x}{4} - 2 = 5$ $\frac{x}{4} = 7 \quad\text{(add 2)}$ $x = 7 \cdot 4 = 28 \quad\text{(multiply by 4)}$

✅ Check: $\dfrac{28}{4} - 2 = 7 - 2 = 5$ ✓

Worked Example: $\dfrac{x}{3} + 6 = 10$

$\frac{x}{3} = 4 \;\Rightarrow\; x = 4 \cdot 3 = 12$

✅ Check: $\dfrac{12}{3} + 6 = 4 + 6 = 10$ ✓

Multiply to Finish 🧮

Solve for $x$ .

1) $\dfrac{x}{2} + 3 = 9 \;\Rightarrow\; x = \,?$ 2) $\dfrac{x}{5} - 1 = 4 \;\Rightarrow\; x = \,?$

Negative Coefficients

A negative number in front of the variable is no problem — just divide by that negative number (a negative divisor flips the sign).

Worked Example: $-3x + 1 = 13$

$-3x + 1 = 13$ $-3x = 12 \quad\text{(subtract 1)}$ $x = \frac{12}{-3} = -4 \quad\text{(divide by }-3)$

✅ Check: $-3(-4) + 1 = 12 + 1 = 13$ ✓

⚠️ Sign care: When you divide by a negative, watch the sign of the result. $\dfrac{12}{-3} = -4$ , not $+4$ .

Constant Written First

Sometimes the constant comes before the variable term, like $b + ax = c$ . The plan is identical — clear the constant, then divide.

Worked Example: $10 - 2x = 4$

$10 - 2x = 4$ $-2x = -6 \quad\text{(subtract 10)}$ $x = \frac{-6}{-2} = 3 \quad\text{(divide by }-2)$

✅ Check: $10 - 2(3) = 10 - 6 = 4$ ✓

💡 The term being subtracted ( $-2x$ ) keeps its negative sign when you isolate it. A negative divided by a negative is positive.

Concept Check 🎯

Spot the Right First Step 🔽

Choose the correct first move for each equation.

Worked Example: $\dfrac{2}{3}x + 4 = 10$

$\frac{2}{3}x + 4 = 10$ $\frac{2}{3}x = 6 \quad\text{(subtract 4)}$ $x = 6 \cdot \frac{3}{2} = \frac{18}{2} = 9 \quad\text{(multiply by reciprocal }\tfrac{3}{2})$

✅ Check: $\dfrac{2}{3}(9) + 4 = 6 + 4 = 10$ ✓

💡 Multiplying by the reciprocal works because $\dfrac{3}{2} \cdot \dfrac{2}{3} = 1$ , leaving $1 \cdot x = x$ .

Decimal Coefficients

Decimals follow the exact same recipe. This shows up constantly in money problems.

Worked Example: $1.5x + 2 = 11$

$1.5x + 2 = 11$ $1.5x = 9 \quad\text{(subtract 2)}$ $x = \frac{9}{1.5} = 6 \quad\text{(divide by 1.5)}$

✅ Check: $1.5(6) + 2 = 9 + 2 = 11$ ✓

Fractions & Decimals 🧮

Solve for $x$ .

1) $\dfrac{3}{4}x - 2 = 7 \;\Rightarrow\; x = \,?$ 2) $0.5x + 3 = 8 \;\Rightarrow\; x = \,?$

Translating Word Problems

Many real situations are two-step equations in disguise. The pattern is usually:

$(\text{rate}) \times (\text{quantity}) + (\text{fixed amount}) = (\text{total})$

Worked Example — The Taxi Ride

A taxi charges a $3 flat fee plus $2 per mile. A ride cost $17. How many miles ($ m$) was it?

$2m + 3 = 17$ $2m = 14 \;\Rightarrow\; m = 7 \text{ miles}$

✅ Check: $2(7) + 3 = 14 + 3 = 17$ dollars ✓

Worked Example — Saving Up

Maria has $20 saved and adds $5 each week. After how many weeks ( $w$ ) will she have $60?

$5w + 20 = 60 \;\Rightarrow\; 5w = 40 \;\Rightarrow\; w = 8 \text{ weeks}$

Word Problem Check 🎯

Set Up & Solve 🧮

A pizza shop charges a $4 delivery fee plus $9 per pizza. The order came to $40.

1) How many pizzas were ordered? $\;p = \,?$

A plant is $6$ cm tall and grows $2$ cm per week.

2) After how many weeks will it be $18$ cm tall? $\;w = \,?$

⚠️ Golden rules: undo addition/subtraction first, do the same thing to both sides, mind your signs, and check by substituting your answer back in.

Mixed Match 🔽

Match each equation to its solution.

Mixed Practice 🧮

Solve each for $x$ .

1) $7x + 8 = 1 \;\Rightarrow\; x = \,?$ 2) $\dfrac{x}{3} - 5 = -2 \;\Rightarrow\; x = \,?$ 3) $\dfrac{2}{5}x + 1 = 7 \;\Rightarrow\; x = \,?$

Application Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.

9(multiply by reciprocal 23)

Solving Two-Step Equations

Solving Two-Step Equations - Complete Interactive Lesson

Part 1: The Big Idea: Undoing Operations

⚖️ Solving Two-Step Equations

Topics in This Part

What Is a Two-Step Equation?

One step vs. two steps

Inverse Operations: The Undo Buttons

The Order: Undo in REVERSE

To solve ax+b=cax + b = cax+b=c:

Part 2: The Standard Move: $ax + b = c$

⚖️ Solving Two-Step Equations

The Two-Step Recipe

Part 3: Division, Negatives & Variable Positions

⚖️ Solving Two-Step Equations

When the Variable Is Divided

Part 4: Fractions, Decimals & Word Problems

⚖️ Solving Two-Step Equations

Fractional Coefficients

Part 5: Mixed Practice & Mastery Check

⚖️ Solving Two-Step Equations

Quick Reference

Worked Example: 3x+7=223x + 7 = 223x+7=22

Worked Example: 5x−4=265x - 4 = 265x−4=26

Worked Example: 2x+9=32x + 9 = 32x+9=3

Worked Example: x4−2=5\dfrac{x}{4} - 2 = 54x​−2=5

Worked Example: x3+6=10\dfrac{x}{3} + 6 = 103x​+6=10

Negative Coefficients

Worked Example: −3x+1=13-3x + 1 = 13−3x+1=13

Constant Written First

Worked Example: 10−2x=410 - 2x = 410−2x=4

Worked Example: 23x+4=10\dfrac{2}{3}x + 4 = 1032​x+4=10

Decimal Coefficients

Worked Example: 1.5x+2=111.5x + 2 = 111.5x+2=11

Translating Word Problems

Worked Example — The Taxi Ride

Worked Example — Saving Up

To solve $ax + b = c$ :

Worked Example: $3x + 7 = 22$

Worked Example: $5x - 4 = 26$

Worked Example: $2x + 9 = 3$

Worked Example: $\dfrac{x}{4} - 2 = 5$

Worked Example: $\dfrac{x}{3} + 6 = 10$

Worked Example: $-3x + 1 = 13$

Worked Example: $10 - 2x = 4$

Worked Example: $\dfrac{2}{3}x + 4 = 10$

Worked Example: $1.5x + 2 = 11$