Solving One and Two-Step Equations

⚖️ Solving One- and Two-Step Equations

Part 1 of 5 — What Is an Equation?

Topics in This Part

🔑 Key Concept: An equation is a statement that two things are equal. To solve it is to find the value of the variable that makes the statement true.

Equations vs. Expressions

An expression is a math phrase with no equals sign — like $x + 4$ or $3n$. You can simplify it, but there's nothing to "solve."

An equation has an equals sign ($=$) joining two expressions:

$x + 4 = 9$

💡 Think of the equals sign as the center of a balance scale. The left side weighs exactly the same as the right side.

Concept Check 🎯

What Does "Solve" Mean?

To solve an equation is to find the number the variable must equal. A solution is correct when, after you substitute it back in, both sides come out the same.

Example: $x + 4 = 9$

Is $x = 5$ a solution? Substitute it in:

$5 + 4 = 9 \quad\Rightarrow\quad 9 = 9 \;\checkmark$

Both sides equal $9$, so yes, $x = 5$ is the solution.

Example: $3n = 12$

Is $n = 4$ a solution?

$3 \cdot 4 = 12 \quad\Rightarrow\quad 12 = 12 \;\checkmark$

Yes — $n = 4$ works.

⚠️ A guess that makes the two sides unequal is not a solution. If $x = 6$ in $x + 4 = 9$, then $6 + 4 = 10 \ne 9$, so $6$ is wrong.

Check the Solution 🎯

Plug In and Check 🧮

For each equation, substitute the given value and write the number the left side equals. (Then you can see whether it matches the right side.)

1) $x + 6 = 11$, try $x = 5$. Left side $= \,?$ 2) $4n = 24$, try $n = 6$. Left side $= \,?$ 3) $y - 8 = 2$, try $y = 10$. Left side $= \,?$

The Balance Idea

Picture an equation as a balanced scale: the left pan weighs the same as the right pan.

$\text{left side} \;=\; \text{right side}$

🔑 The Golden Rule of Equations: Whatever you do to one side, you must do to the other side — exactly. That keeps the scale balanced, so the equation stays true.

If you add $3$ to the left, add $3$ to the right. If you divide the left by $2$, divide the right by $2$. This single rule is the engine behind every technique in this lesson — and it's what Part 2 puts to work.

⚖️ Solving One- and Two-Step Equations

Part 2 of 5 — Inverse Operations & One-Step Equations

🔑 The Idea: To get the variable alone, undo whatever is being done to it. Use the opposite operation — and do it to both sides.

Operations and Their Inverses

Every operation has an inverse that undoes it:

To solve a one-step equation, ask: "What is being done to the variable?" Then do the opposite to both sides.

💡 The goal every time is to isolate the variable — get it alone on one side, with its coefficient equal to $1$.

Undoing Addition and Subtraction

Example: $x + 7 = 12$

The $7$ is being added, so subtract $7$ from both sides:

$x + 7 - 7 = 12 - 7$ $x = 5$

✅ Check: $5 + 7 = 12$ ✓

Example: $x - 4 = 9$

The $4$ is being subtracted, so add $4$ to both sides:

$x - 4 + 4 = 9 + 4$ $x = 13$

✅ Check: $13 - 4 = 9$ ✓

Concept Check 🎯

Undoing Multiplication and Division

Example: $5x = 30$

Here $5x$ means $5 \cdot x$ — the variable is being multiplied by $5$. Undo it by dividing both sides by $5$:

$\frac{5x}{5} = \frac{30}{5}$ $x = 6$

✅ Check: $5 \cdot 6 = 30$ ✓

Example: $\dfrac{x}{3} = 4$

Here $x$ is being divided by $3$. Undo it by multiplying both sides by $3$:

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

✅ Check: $\dfrac{12}{3} = 4$ ✓

⚠️ Watch the notation: $5x$ means "multiply," so you divide to undo it. $\dfrac{x}{3}$ means "divide," so you multiply to undo it.

Pick the Inverse Move 🔽

For each equation, choose the operation that isolates the variable in one step.

Solve One-Step Equations 🧮

Solve for the variable. Enter just the number.

1) $x + 13 = 20 \;\Rightarrow\; x = \,?$ 2) $8x = 56 \;\Rightarrow\; x = \,?$ 3) $\dfrac{x}{5} = 7 \;\Rightarrow\; x = \,?$ 4) $x - 6 = 10 \;\Rightarrow\; x = \,?$

⚖️ Solving One- and Two-Step Equations

Part 3 of 5 — Two-Step Equations

🔑 The Idea: A two-step equation has two operations on the variable. Undo them in reverse order: first undo addition/subtraction, then undo multiplication/division.

Undo in Reverse Order

A two-step equation like $2x + 3 = 11$ does two things to $x$:

1. Multiplies by $2$
2. Adds $3$

To undo, work backwards — like taking off your shoes before your socks:

🔑 Reverse-Order Rule: First undo what's farthest from the variable (the $+$ or $-$ constant). Then undo the multiplication or division.

So for $2x + 3 = 11$: first subtract $3$, then divide by $2$.

💡 This is the opposite of the order of operations. When building an expression you multiply then add; when solving you subtract then divide.

Worked Example: $2x + 3 = 11$

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

$2x + 3 - 3 = 11 - 3$ $2x = 8$

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

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

✅ Check: $2 \cdot 4 + 3 = 8 + 3 = 11$ ✓

Worked Example: $\dfrac{x}{3} - 5 = 1$

Step 1 — Undo the $-5$. Add $5$ to both sides:

$\frac{x}{3} - 5 + 5 = 1 + 5$ $\frac{x}{3} = 6$

Step 2 — Undo the $\div 3$. Multiply both sides by $3$:

$3 \cdot \frac{x}{3} = 3 \cdot 6$ $x = 18$

✅ Check: $\dfrac{18}{3} - 5 = 6 - 5 = 1$ ✓

Order the Steps 🔽

You're solving $5x - 4 = 16$. Choose what happens at each stage.

Concept Check 🎯

Solve Two-Step Equations 🧮

Solve for the variable. Enter just the number.

1) $2x + 5 = 17 \;\Rightarrow\; x = \,?$ 2) $3x - 4 = 20 \;\Rightarrow\; x = \,?$ 3) $\dfrac{x}{4} + 2 = 6 \;\Rightarrow\; x = \,?$

⚖️ Solving One- and Two-Step Equations

Part 4 of 5 — Negatives, Fractions & Word Problems

🔑 The Idea: The same two rules — use inverses and do it to both sides — handle negatives, fraction coefficients, and real-world problems. Only the numbers change.

Equations with Negative Numbers

The rules don't change with negatives — just track the signs carefully.

Example: $x + 9 = 4$

Subtract $9$ from both sides:

$x = 4 - 9 = -5$

✅ Check: $-5 + 9 = 4$ ✓

Example: $-3x = 21$

The variable is multiplied by $-3$, so divide both sides by $-3$:

$x = \frac{21}{-3} = -7$

✅ Check: $-3 \cdot (-7) = 21$ ✓

⚠️ Dividing by a negative flips the sign. A positive divided by a negative is negative.

Concept Check 🎯

Fraction Coefficients

When the variable is multiplied by a fraction, multiply both sides by that fraction's reciprocal (flip it).

Example: $\dfrac{2}{3}x = 8$

The reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$. Multiply both sides by $\dfrac{3}{2}$:

$\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 8$ $x = \frac{3 \cdot 8}{2} = \frac{24}{2} = 12$

✅ Check: $\dfrac{2}{3} \cdot 12 = \dfrac{24}{3} = 8$ ✓

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

Turning Words into Equations

Word problems become solvable once you translate them. Watch for these signal words:

Example

"Five more than twice a number is $17$. Find the number."

"Twice a number" is $2x$; "five more than" that is $2x + 5$; "is $17$" gives $= 17$:

$2x + 5 = 17 \;\Rightarrow\; 2x = 12 \;\Rightarrow\; x = 6$

The number is $\mathbf{6}$.

⚠️ "Less than" reverses the order. "$3$ less than a number" is $x - 3$, not $3 - x$.

Translate the Words 🔽

Choose the equation that matches each sentence. ($n$ = the unknown number.)

Solve It 🧮

Solve each. Negative answers and fractions are fine (write fractions like $-5/2$).

1) $x + 15 = 8 \;\Rightarrow\; x = \,?$ 2) $\dfrac{3}{4}x = 9 \;\Rightarrow\; x = \,?$ 3) "Six less than a number is $10$." The number is $\,?$

⚖️ Solving One- and Two-Step Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) check solutions, (2) solve one-step equations with inverses, (3) solve two-step equations in reverse order, and (4) handle negatives, fractions, and word problems. Let's put it all together.

Quick Reference

🔑 Two golden rules: Use the inverse operation, and do the same thing to both sides. For two-step equations, undo the $+/-$ before the $\times/\div$.

⚠️ Always check by substituting your answer back into the original equation.

Mixed Practice 🎯

Final Drill 🧮

Solve each. Enter just the number.

1) $x - 14 = 6 \;\Rightarrow\; x = \,?$ 2) $9x = 72 \;\Rightarrow\; x = \,?$ 3) $5x + 8 = 43 \;\Rightarrow\; x = \,?$ 4) $-6x = 30 \;\Rightarrow\; x = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.