Expressions and Equations: Chapter 3

• The following is a checklist of skills covered in Chapter 3, "Expressions and equations"

Algebraic Expressions

• An algebraic expression consists of terms, coefficients, and constant terms.
• Example: In the expression 3x − 5y + 6, there are 3 terms.
• The coefficient of y is -5.
• The constant term is 6.

Converting Words to Expressions

• Algebraic expressions can represent word problems.
• For example, the cost of n movie tickets at $15 each can be expressed as 15n.
• "Four less than three times x" is written as 3x − 4.
• "The square of y is divided by 4" translates to y²/4.

Evaluating Expressions

• Substitute given values into expressions to evaluate them.
• Example: If p = −4, q = 6, and r = 5, then 4q − r = 4(6) − 5 = 19.
• 3p − (q + r) = 3(−4) − (6 + 5) = −12 − 11 = −23.

Identifying Like Terms

• Like terms have the same variable raised to the same power.
• From the set 4x, 5, 3xy, −5x, 2xy², yx, the pairs of like terms are 4x and −5x, and 3xy and yx.

Collecting Like Terms

• Simplify expressions by combining like terms.
• Example: 5a − 4 + 3a = 8a − 4
• 4xy + 6y − xy + 4y = 3xy + 10y

Multiplying Algebraic Terms

• Simplify expressions by multiplying algebraic terms.
• Example: 4s × 5t = 20st
• −3y × 8xy = −24xy²

Dividing Algebraic Terms

• Simplify expressions by dividing algebraic terms.
• Example: (15x)/12 = 5x/4
• (16a²b) ÷ (24ab) = (2a)/3

Expanding Expressions with Brackets

• Expand expressions and simplify
• Example: 4(x + 5) = 4x + 20
• −3(x − 6) = −3x + 18
• 2x(3x − y) = 6x² − 2xy

Expanding and Collecting Like Terms

• Expand and then simplify expressions by collecting like terms.
• Example: 3(x + 2) − 4 = 3x + 6 − 4 = 3x + 2
• 5 − 2(x + 1) = 5 − 2x − 2 = 3 − 2x

Solving Simple Linear Equations

• Solve for the variable in linear equations.
• Example: 2x − 5 = 7 => 2x = 12 => x = 6
• 8 − 3x = 15 => −3x = 7 => x = −7/3

Solving Linear Equations with Fractional Coefficients

• Solve equations involving fractions.
• Example: (3x)/4 = 9 => 3x = 36 => x = 12
• x/5 − 4 = 2 => x/5 = 6 => x = 30

Solving Simple Fractional Equations

• Solve fractional equations for the unknown variable.
• Example: (x − 5)/3 = 6 => x − 5 = 18 => x = 23

Solving Equations with Brackets

• Simplify and solve equations containing brackets.
• Example: 3(3x + 5) = 22 => 9x + 15 = 22 => 9x = 7 => x = 7/9

Solving Equations with Brackets and a Common Factor

• Solve equations by first dividing by a common factor.
• Example: 4(2x + 3) = 32 => 2x + 3 = 8 => 2x = 5 => x = 5/2

Solving Equations with Brackets and Like Terms

• Expand, simplify, and solve equations with brackets and like terms.
• Example: 2(4x + 7) − 5x = 20 => 8x + 14 − 5x = 20 => 3x = 6 => x = 2

Solving Equations with Pronumerals on Both Sides

• Rearrange and solve equations with variables on both sides.
• Example: 8x = 3x + 25 => 5x = 25 => x = 5
• 10 − 3x = x + 2 => 8 = 4x => x = 2

Solving Equations with Brackets and Pronumerals on Both Sides

• Expand, simplify, and solve equations with brackets and variables on both sides.
• Example: 4(2x − 1) = 2(x − 8) => 8x − 4 = 2x − 16 => 6x = −12 => x = −2

Turning Word Problems into Equations

• Convert word problems into algebraic equations.
• Example: "8 more than two times a certain number is 30" can be written as 2x + 8 = 30. Solving for x gives x = 11.

Applying Algebra to Word Problems

• Apply algebraic concepts to solve practical word problems.
• Example: A surf shop charges $15 + $12/hour. If Jett is charged $51, then 15 + 12h = 51 => 12h = 36 => h = 3 hours.

Solving More Complex Word Problems

• Use algebra to solve complex real-world problems.
• Example: Toni and Trey earn $370. Toni earns $140 more than Trey. If Trey earns T, then Toni earns T + 140. T + (T + 140) = 370 => 2T + 140 = 370 => 2T = 230 => T = $115. Toni earns $255.

Substituting Values into Formulas

• Substitute given values into formulas to find unknowns.
• Example: A = (1/2)xy, when x = 7 and y = 10, A = (1/2)(7)(10) = 35.
• V = 휋r²h, when V = 120 and r = 3, 120 = 휋(3²)(h) => h = 120/(9휋) ≈ 4.2.

Transposing a Formula (Extension)

• Rearrange a formula to make a different variable the subject.
• Example: v² = u² + 2as, solving for u (u > 0): u² = v² - 2as => u = √(v² - 2as).