Lesson 2: Solve Two-Step Equations

Property

To solve an equation, the goal is to determine the value of the unknown.
This is done by using inverse operations to isolate the variable.
For an equation like $ax + b = c$, first use the addition or subtraction property of equality to isolate the term with the variable ($ax$).
Then, use the multiplication or division property of equality to find the value of $x$.

Examples

• To solve $4x + 5 = 25$, first subtract 5 from both sides to get $4x = 20$. Then, divide both sides by 4 to find $x = 5$.
• To solve $3p - 8 = 13$, first add 8 to both sides to get $3p = 21$. Then, divide both sides by 3 to find $p = 7$.
• To solve $\frac{k}{3} + 2 = 6$, first subtract 2 from both sides to get $\frac{k}{3} = 4$. Then, multiply both sides by 3 to find $k = 12$.

Explanation

Think of solving an equation as unwrapping a present. You must undo the operations in reverse order to find the variable. First, undo any addition or subtraction, then undo any multiplication or division to get the variable by itself.

Property

A tape diagram models a word problem by representing a total quantity as a whole bar and its parts as sections of the bar.
The visual relationship between the parts and the whole helps determine the sequence of operations (addition, subtraction, multiplication, division) needed to find the unknown value, $x$.

Examples

Property

A problem can be solved arithmetically by working backward or algebraically by writing and solving an equation. Both methods use inverse operations to find the unknown value. For a problem modeled by $ax + b = c$, the arithmetic approach undoes the addition/subtraction first, then the multiplication/division, just as the algebraic approach does.

Examples

• Problem: A taxi ride costs a flat fee of 3 dollars plus 2 dollars per mile. If a ride costs 19 dollars, how many miles was it?

Arithmetic Solution: Start with the total cost of 19 dollars. Subtract the flat fee: $19 - 3 = 16$ dollars. Divide by the cost per mile: $16 \div 2 = 8$ miles.
Algebraic Solution: Let $m$ be the number of miles. The equation is $2m + 3 = 19$. Subtract 3 from both sides: $2m = 16$. Divide by 2: $m = 8$.

• Problem: Sarah sold 4 equal-sized boxes of cookies, but first she ate 5 cookies. In total, she sold 43 cookies. How many cookies were in each full box?

Arithmetic Solution: Start with the 43 cookies sold. Add back the 5 cookies she ate: $43 + 5 = 48$. Divide by the 4 boxes: $48 \div 4 = 12$ cookies per box.
Algebraic Solution: Let $c$ be the cookies in a box. The equation is $4c - 5 = 43$. Add 5 to both sides: $4c = 48$. Divide by 4: $c = 12$.

Explanation

The arithmetic method involves "working backward" from the final result, step-by-step, using inverse operations. The algebraic method involves representing the unknown with a variable, writing an equation, and then using properties of equality to solve. Both approaches rely on the same logic of undoing operations to isolate the unknown quantity. Understanding both methods helps connect your intuitive problem-solving skills to the formal process of algebra.