Lesson 4: More Equations

New Concept

This lesson introduces a powerful strategy for solving equations with multiple steps. We'll learn to isolate the variable by "unwrapping" it—undoing each operation in reverse order to find the solution.

What’s next

You'll soon tackle interactive examples and practice cards that apply this strategy to solve equations, formulas, and real-world problems.

Property

To solve an equation with two or more operations, we must isolate the variable on one side of the equation. We undo the operations in reverse order. Typically, we undo addition or subtraction first, before undoing multiplication or division.

Examples

• To solve $4x + 5 = 29$, first subtract 5 from both sides to get $4x = 24$. Then, divide both sides by 4 to find $x = 6$.
• To solve $\frac{y}{3} - 2 = 7$, first add 2 to both sides to get $\frac{y}{3} = 9$. Then, multiply both sides by 3 to find $y = 27$.
• To solve $18 = 6 + 2z$, first subtract 6 from both sides to get $12 = 2z$. Then, divide both sides by 2 to find $z = 6$.

Explanation

Think of it as reversing your morning routine. To get back to the start, you undo the last thing you did first. In equations, this means handling addition or subtraction before dealing with multiplication or division to isolate the variable.

Property

To solve an equation:

1. List the operations performed on the variable in order.
2. Undo those operations in reverse order.

Examples

• Solve $\frac{4a - 5}{3} = 9$. The operations on $a$ are: multiply by 4, subtract 5, divide by 3. Undo in reverse: multiply by 3 to get $4a - 5 = 27$, add 5 to get $4a = 32$, and divide by 4 to get $a = 8$.
• Solve $\frac{b + 7}{2} - 3 = 8$. Undo in reverse: add 3 to get $\frac{b+7}{2} = 11$, multiply by 2 to get $b+7 = 22$, and subtract 7 to get $b = 15$.
• Solve $10 + \frac{c}{5} = 14$. The operations on $c$ are: divide by 5, then add 10. Undo in reverse: subtract 10 to get $\frac{c}{5} = 4$, then multiply by 5 to get $c = 20$.

Explanation

Solving an equation is like unwrapping a gift. The variable is the gift inside, and the operations are the wrapping paper and box. To get to the gift, you must undo each layer in the reverse order it was applied.

Property

A formula is an equation that describes a relationship between variables. We use formulas to solve for an unknown value by substituting the known values into the formula and then solving the resulting equation.

Examples

• The perimeter of a rectangle is $P = 2l + 2w$. If $P=60$ cm and $w=12$ cm, find the length $l$. Substitute to get $60 = 2l + 2(12)$, which is $60 = 2l + 24$. Solving gives $36 = 2l$, so $l=18$ cm.
• Using the temperature formula $F = \frac{9}{5}C + 32$, find the Celsius temperature for 59°F. Substitute to get $59 = \frac{9}{5}C + 32$. Subtract 32 to get $27 = \frac{9}{5}C$. Solving for $C$ gives $C = 15$ degrees.
• The area of a triangle is $A = \frac{1}{2}bh$. If $A=50$ square inches and $b=10$ inches, find the height $h$. Substitute to get $50 = \frac{1}{2}(10)h$, which is $50 = 5h$. Solving gives $h=10$ inches.

Explanation

Formulas are like pre-built equations for real-world situations. Just plug in the numbers you know, and the formula turns into a simple equation puzzle. Solve it to find the missing piece of information you need.

Property

Steps for Modeling a Problem:

1. Identify the unknown quantity and choose a variable to represent it.
2. Find some quantity that can be expressed in two different ways, and write an equation.
3. Solve the equation, and answer the question in the problem.

Examples

• A taxi ride costs 3 dollars plus 2 dollars per mile. If the total fare was 21 dollars, how many miles was the ride? Let $m$ be the miles. The equation is $3 + 2m = 21$. Solving gives $2m = 18$, so the ride was $m=9$ miles.
• Jamal is saving for a 150 dollars bike. He has 30 dollars and earns 15 dollars per week. How many weeks will it take? Let $w$ be the weeks. The equation is $30 + 15w = 150$. Solving gives $15w = 120$, so it will take $w=8$ weeks.
• A phone company charges 20 dollars a month plus 0.05 dollars per text. If a monthly bill is 24 dollars, how many texts were sent? Let $t$ be the number of texts. The equation is $20 + 0.05t = 24$. Solving gives $0.05t = 4$, so $t=80$ texts were sent.

Explanation

Algebra is a powerful tool that turns word problems into solvable puzzles. First, you pick a letter for the unknown value. Then, you translate the problem's words into a math sentence (an equation) and solve it to find the answer.