Unit 2 Review: Linear Equations, Ratios, and Proportions

Review of Linear Equations, Ratios, and Proportions

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the first power. Solving these equations is a foundational skill in algebra.

• Definition: A linear equation in one variable has the form $ax + b = 0$, where $a \neq 0$.

• Steps to Solve:

  1. Isolate the variable on one side of the equation.

  2. Use inverse operations (addition, subtraction, multiplication, division) to solve for the variable.

• Example: Solve $2x + 3 = 7$.

  • Subtract 3 from both sides: $2x = 4$

  • Divide both sides by 2: $x = 2$

Applications: Translating and Solving Word Problems

Many real-world problems can be modeled and solved using linear equations.

• Key Steps:

  1. Define the variable(s).

  2. Translate the problem into an equation.

  3. Solve the equation.

  4. Interpret the solution in the context of the problem.

• Example: If a number increased by 5 is 12, what is the number?

  • Let $x$ be the number. $x + 5 = 12$

  • $x = 7$

Properties of Equality and Solving Equations

Understanding the properties of equality is essential for manipulating and solving equations.

• Addition Property: If $a = b$, then $a + c = b + c$.

• Multiplication Property: If $a = b$, then $ac = bc$.

• Application: These properties allow you to perform the same operation on both sides of an equation to maintain equality.

Ratios and Proportions

Ratios compare two quantities, while proportions state that two ratios are equal.

• Ratio: A comparison of two numbers by division, written as $a:b$ or $\frac{a}{b}$.

• Proportion: An equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$.

• Solving Proportions: Use cross-multiplication: $a \times d = b \times c$.

• Example: Solve $\frac{2}{3} = \frac{x}{6}$.

  • Cross-multiply: $2 \times 6 = 3 \times x$

  • $12 = 3x$

  • $x = 4$

Applications of Ratios and Proportions

Ratios and proportions are used in scaling, map reading, recipe adjustments, and more.

• Example: If a map scale is 1 inch : 5 miles, how many miles does 3 inches represent?

  • Set up a proportion: $\frac{1}{5} = \frac{3}{x}$

  • Cross-multiply: $1 \times x = 5 \times 3$

  • $x = 15$ miles

Graphing Linear Equations

Graphing is a visual way to represent solutions to linear equations in two variables.

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Graphing Steps:

  1. Find the y-intercept ($b$).

  2. Use the slope ($m$) to find another point.

  3. Draw a straight line through the points.

• Example: Graph $y = 2x + 1$.

  • y-intercept: (0, 1)

  • Slope: 2 (rise 2, run 1)

Types of Equations and Solutions

Linear equations can have one solution, no solution, or infinitely many solutions.

• One Solution: The equation simplifies to $x = a$.

• No Solution: Contradictory statement, e.g., $0 = 5$.

• Infinitely Many Solutions: Identity, e.g., $0 = 0$ after simplification.

Formulas and Applications

Many problems require the use of formulas, such as those for perimeter, area, or distance.

• Distance Formula: $d = rt$ (distance = rate × time)

• Perimeter of a Rectangle: $P = 2l + 2w$

• Area of a Rectangle: $A = lw$

Summary Table: Types of Linear Equations

Type	Form	Number of Solutions
Consistent & Independent	Distinct lines	One
Inconsistent	Parallel lines	None
Dependent	Same line	Infinitely many

Additional info:

• This review covers Sections 2.1 – 2.7 and a supplement on ratios and proportions, which are foundational topics in beginning-intermediate algebra.

• Students should be able to apply these concepts to solve equations, interpret word problems, and graph solutions.