2.1 Use a General Strategy to Solve Linear Equations

Linear equations are the building blocks of algebra. They're everywhere, from calculating your grocery bill to figuring out how long it'll take to get to your friend's house. Mastering these equations is key to unlocking more complex math concepts.

In this section, we'll break down the steps to solve linear equations. We'll cover different types, tackle fractions and decimals, and explore equivalent equations. By the end, you'll be solving linear equations like a pro!

Solving Linear Equations

Steps for solving linear equations

• Simplify each side of the equation
  • Combine like terms on each side of the equation by adding or subtracting coefficients of variables with the same exponent ($3x + 2x = 5x$)
  • Use the distributive property to remove parentheses by multiplying the term outside the parentheses by each term inside ($2(3x + 4) = 6x + 8$)
• Isolate the variable term on one side of the equation
  • Add or subtract terms from both sides to get the variable term alone on one side (if the equation has $2x - 3 = 7$, add 3 to both sides to get $2x = 10$)
  • Combine constant terms on the opposite side of the equation from the variable term
  • Use inverse operations to undo operations on the variable term
• Divide both sides by the coefficient of the variable term to solve for the variable
  • After isolating the variable term, divide both sides by its coefficient to get the variable alone ($2x = 10$ becomes $x = 5$ by dividing both sides by 2)
  • Check the solution by substituting the value back into the original equation to ensure it creates a true statement

Types of linear equations

• Identify the number of solutions based on the equation's structure
  • One solution: The variable term can be isolated, and the equation has a single solution ($2x - 3 = 7$ has one solution, $x = 5$)
  • No solution: The equation simplifies to a false statement, such as $0 = 1$ (an example is $2x - 2x = 1$, which simplifies to $0 = 1$)
  • Infinitely many solutions: The equation simplifies to a true statement, such as $0 = 0$ ($2x - 2x = 0$ simplifies to $0 = 0$, so any value of x is a solution)
• Recognize equations with variables on both sides
  • Collect variable terms on one side and constant terms on the other side by adding or subtracting terms from both sides ($2x + 3 = x + 7$ becomes $x = 4$ by subtracting x from both sides and then subtracting 3 from both sides)
  • Combine like terms on each side and solve for the variable
• Identify equations with parentheses and use the distributive property
  • Multiply the term outside the parentheses by each term inside the parentheses ($2(3x + 4) = 10$ becomes $6x + 8 = 10$)
  • Combine like terms and solve for the variable using the steps for isolating the variable and dividing by the coefficient

Solving equations with fractions

• Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate fractions
  1. Find the LCM of all denominators in the equation
  2. Multiply each term on both sides by the LCM to eliminate fractions ($\frac{x}{2} + \frac{x}{3} = 5$ becomes $3x + 2x = 30$ by multiplying both sides by the LCM of 6)
• Convert decimals to fractions and then multiply by the LCM
  • Write each decimal as a fraction with a denominator of a power of 10 (0.5 becomes $\frac{5}{10}$ and 0.25 becomes $\frac{25}{100}$)
  • Find the LCM of the denominators and multiply both sides by it to eliminate decimals and fractions
• Solve the resulting equation using the step-by-step approach
  1. Combine like terms
  2. Isolate the variable by adding or subtracting terms from both sides
  3. Divide by the coefficient of the variable term
  4. Check the solution by substituting it back into the original equation with fractions or decimals to ensure it creates a true statement

Understanding Equivalent Equations and Algebraic Expressions

• Recognize that equivalent equations have the same solution set
  • Equivalent equations are formed by performing the same operation on both sides of the equation
  • The identity property of equality states that adding or subtracting the same value from both sides of an equation results in an equivalent equation
• Work with algebraic expressions to simplify equations
  • Algebraic expressions are mathematical phrases that contain variables, numbers, and operations
  • Simplifying algebraic expressions can help in solving linear equations more efficiently