Solving Two-Step Equations

What Is a Two-Step Equation? 🧩

A two-step equation is an equation that needs two different operations to solve.

You already know how to solve a one-step equation like $x + 5 = 17$ (just subtract 5). A two-step equation has one extra layer:

$3x + 5 = 17$

To get $x$ all by itself here, you have to undo two things:

• the $+ 5$ that is added to $3x$, and
• the $3$ that is multiplied by $x$.

The whole goal is to isolate the variable — to get $x$ alone on one side of the equals sign so the other side tells you its value.

The Golden Rule: Stay Balanced ⚖️

Think of an equation as a balanced scale. The two sides are equal, so whatever you do to one side, you must do to the other side to keep it balanced.

We solve by using inverse (opposite) operations to peel away everything around the variable:

Operation you see	Inverse used to undo it
addition ($+$)	subtraction ($-$)
subtraction ($-$)	addition ($+$)
multiplication ($\times$)	division ($\div$)
division ($\div$)	multiplication ($\times$)

Order matters. Just like getting dressed, you take things off in the reverse order you put them on.

The Two Steps, In Order 🪜

For an equation like $3x + 5 = 17$:

Step 1 — Undo the addition or subtraction first. Move the constant (the lonely number) to the other side. $3x + 5 - 5 = 17 - 5 \;\Rightarrow\; 3x = 12$

Step 2 — Undo the multiplication or division second. Now isolate the variable. $\frac{3x}{3} = \frac{12}{3} \;\Rightarrow\; x = 4$

Why this order? The variable is "wrapped" by multiplication inside and addition outside. We unwrap from the outside in — strip the added number off first, then the number multiplying the variable. Doing it in this order keeps the arithmetic clean.

Quick Concept Check ✅

Make sure the big idea is solid before we start solving.

Worked Example 1: $3x + 5 = 17$ ✏️

Let's solve it slowly, one line at a time.

Start: $3x + 5 = 17$

Step 1 — Subtract 5 from both sides (undo the $+5$): $3x + 5 - 5 = 17 - 5$ $3x = 12$

Step 2 — Divide both sides by 3 (undo the $\times 3$): $\frac{3x}{3} = \frac{12}{3}$ $x = 4$

Check your answer — substitute $x = 4$ back into the original equation: $3(4) + 5 = 12 + 5 = 17 ✓$

It works, so $x = 4$ is correct. Always check — it catches almost every mistake.

Worked Example 2: Working With Negatives ➖

Negatives follow the exact same two steps. Just be careful with signs.

Solve: $-2x - 6 = 10$

Step 1 — Add 6 to both sides (undo the $-6$): $-2x - 6 + 6 = 10 + 6$ $-2x = 16$

Step 2 — Divide both sides by $-2$ (undo the $\times -2$): $\frac{-2x}{-2} = \frac{16}{-2}$ $x = -8$

Check: $-2(-8) - 6 = 16 - 6 = 10 ✓$

Notice that dividing a positive ($16$) by a negative ($-2$) gives a negative answer. Watch those signs in Step 2!

Your Turn: Fill In the Steps 🔧

Solve $5x + 3 = 23$ by filling in each blank with a number.

• Box 1: After subtracting 3 from both sides, $5x = \;?$
• Box 2: After dividing both sides by 5, $x = \;?$
• Box 3: Check it — what is $5x + 3$ when you plug your answer back in?

Guided Practice: Choose the Solution 🎯

Work each one on paper using the two steps, then pick the answer.

Pick the Right Step 🧠

For the equation $\frac{x}{3} + 4 = 10$, choose the correct first step and the correct final answer.

Two-Step Equations in Real Life 🌎

Two-step equations show up constantly once you start charging fees, splitting bills, or saving money.

Example — Skating rink: A roller rink charges a flat $5 entry fee plus $3 per hour of skating. You spent $17 total. How many hours did you skate?

Set it up. Let $h$ be the number of hours: $3h + 5 = 17$

That $5 entry is the constant, and the $3-per-hour is the rate multiplying the variable — a classic two-step setup!

Solve it.

• Step 1: subtract 5 → $3h = 12$
• Step 2: divide by 3 → $h = 4$

You skated for 4 hours. The skill is the same as before — the only new part is translating words into an equation first.

Application: Saving for a Bike 🚲

Maria has $12 saved. She adds $8 every week. She needs $60 for a bike. The equation is $8w + 12 = 60$, where $w$ is the number of weeks.

• Box 1: After subtracting 12 from both sides, $8w = \;?$
• Box 2: After dividing both sides by 8, $w = \;?$ (this is the number of weeks)

Application Check 🧾

Read carefully and translate the words into a two-step equation.

Putting It All Together 🏁

You now have a reliable, two-step recipe for any equation in the form $ax + b = c$.

Step	What you do	Why
1	Undo addition/subtraction	Move the constant off the variable's side
2	Undo multiplication/division	Isolate the variable completely
3	Check by substituting	Confirm both sides are equal

Remember:

• Keep the equation balanced — do the same thing to both sides.
• Watch your signs, especially when dividing by a negative.
• Always check your answer in the original equation.

Time for a final mixed challenge that brings together positives, negatives, and division. You've got this! 💪

Final Challenge 🎓

Two mixed problems — take your time and check each answer.