📘Intermediate Algebra Unit 2 – Solving Linear Equations

What's This All About?

• Linear equations represent a straight line on a graph and have the general form $ax + b = c$, where $a$, $b$, and $c$ are constants and $x$ is the variable
• Solving linear equations involves finding the value of the variable that makes the equation true
• Linear equations can have one solution ($x = 3$), no solution ($0 = 1$), or infinitely many solutions ($2x = 2x$)
• Solving linear equations is a fundamental skill in algebra and is used in various fields, such as physics, engineering, and economics
• Understanding how to solve linear equations is essential for more advanced topics in algebra, such as systems of equations and inequalities

Key Concepts You Need to Know

• Variables represent unknown values in an equation, usually denoted by letters like $x$, $y$, or $z$
• Coefficients are the numbers that multiply the variables in an equation ($2x$, where $2$ is the coefficient)
• Constants are the numbers in an equation that are not multiplied by a variable ($x + 3$, where $3$ is the constant)
• Like terms are terms that have the same variable raised to the same power ($2x$ and $3x$ are like terms)
• The equality sign ($=$) indicates that the expressions on both sides of the equation have the same value
• Inverse operations are used to isolate the variable and solve the equation (addition and subtraction, multiplication and division)
  • To undo addition, subtract the same value from both sides of the equation
  • To undo subtraction, add the same value to both sides of the equation
  • To undo multiplication, divide both sides of the equation by the same value
  • To undo division, multiply both sides of the equation by the same value

Breaking Down Linear Equations

• Identify the variable, coefficients, and constants in the equation
• Combine like terms on each side of the equation by adding or subtracting their coefficients ($2x + 3x = 5x$)
• Use the distributive property to remove parentheses when necessary ($2(x + 3) = 2x + 6$)
• Isolate the variable by performing the same operation on both sides of the equation
  • Add or subtract constants to get the variable terms on one side and the constant terms on the other side ($2x + 3 = 7$ becomes $2x = 4$)
  • Divide both sides by the coefficient of the variable to solve for the variable ($2x = 4$ becomes $x = 2$)
• Check your solution by substituting the value back into the original equation to ensure it makes the equation true

Step-by-Step Solving Methods

• Simplify each side of the equation by combining like terms and using the distributive property
• Add or subtract constants to isolate the variable terms on one side of the equation
• Add or subtract variable terms to get all variable terms on one side of the equation
• Divide both sides of the equation by the coefficient of the variable to solve for the variable
• Check your solution by substituting the value back into the original equation
• Example: Solve $2(3x - 1) + 4x = 5x + 8$
  • Simplify: $6x - 2 + 4x = 5x + 8$
  • Combine like terms: $10x - 2 = 5x + 8$
  • Subtract $5x$ from both sides: $5x - 2 = 8$
  • Add $2$ to both sides: $5x = 10$
  • Divide both sides by $5$: $x = 2$
  • Check: $2(3(2) - 1) + 4(2) = 5(2) + 8$ simplifies to $14 = 14$, so the solution is correct

Common Pitfalls and How to Avoid Them

• Forgetting to distribute when removing parentheses can lead to incorrect simplification
  • Always use the distributive property when removing parentheses ($2(x + 3) = 2x + 6$, not $2x + 3$)
• Subtracting a negative number is the same as adding a positive number
  • Be careful with signs when subtracting negative terms ($x - (-3) = x + 3$)
• Dividing by a negative coefficient will change the direction of the inequality sign
  • When solving inequalities, if you divide by a negative number, reverse the inequality sign ($-2x < 6$ becomes $x > -3$)
• Checking your solution is crucial to ensure you haven't made any mistakes along the way
  • Always substitute your solution back into the original equation to verify its correctness

Real-World Applications

• Calculating the cost of items based on quantity and price per unit ($5x + 10 = 60$, where $x$ is the number of items and $60$ is the total cost)
• Determining the time required to complete a task based on the rate of work and the amount of work done ($2x + 3 = 15$, where $x$ is the number of hours worked)
• Finding the break-even point in a business, where revenue equals costs ($500x = 2000 + 100x$, where $x$ is the number of units sold)
• Balancing chemical equations in chemistry ($2H_2 + O_2 = 2H_2O$)
• Solving for an unknown dimension in geometry problems ($2x + 3 = 15$, where $x$ is the length of a side of a rectangle)

Practice Problems and Tips

• Practice solving linear equations with various levels of difficulty to build your skills and confidence
• Start with simple equations and gradually progress to more complex ones
• Pay attention to the signs of the terms and coefficients, as they can affect the direction of the inequality sign when solving inequalities
• Double-check your work by substituting your solution back into the original equation
• Example: Solve $3(2x - 1) - 4(x + 2) = 5x - 7$
  • Simplify: $6x - 3 - 4x - 8 = 5x - 7$
  • Combine like terms: $2x - 11 = 5x - 7$
  • Subtract $2x$ from both sides: $-11 = 3x - 7$
  • Add $7$ to both sides: $-4 = 3x$
  • Divide both sides by $3$: $-\frac{4}{3} = x$
  • Check: $3(2(-\frac{4}{3}) - 1) - 4((-\frac{4}{3}) + 2) = 5(-\frac{4}{3}) - 7$ simplifies to $-7 = -7$, so the solution is correct

Beyond Basic Linear Equations

• Linear equations can be extended to include absolute value expressions ($|2x - 3| = 7$)
• Systems of linear equations involve solving two or more linear equations simultaneously to find the values of multiple variables
• Inequalities are similar to equations but use inequality signs ($<$, $>$, $\leq$, $\geq$) instead of the equality sign ($=$)
• Graphing linear equations and inequalities can help visualize the solution sets and understand the relationships between variables
• More advanced topics in algebra, such as quadratic equations and exponential functions, build upon the foundational skills of solving linear equations