Solving Two-Step Equations

⚖️ Solving Two-Step Equations

Part 1 of 5 — The Big Idea: Undoing Operations

Topics in This Part

🔑 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

💡 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.

🔑 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. ⚖️

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 $ax + b = c$:

1. First undo the $+b$ or $-b$ (the addition/subtraction). ➡️ clears the loose number
2. 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.

Order Check 🎯

⚖️ 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 $ax + b = c$:

1. Subtract $b$ from both sides → $ax = c - b$
2. Divide by $a$ on both sides → $x = \dfrac{c - b}{a}$
3. 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) $5x + 12 = 2 \;\Rightarrow\; x = \,?$

⚖️ 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 divided by $a$, so you undo it by multiplying by $a$.

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.

⚖️ 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).

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? $\;w = \,?$

⚖️ 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

⚠️ 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.