Solving Linear Equations in Algebra

Solving Linear Equations

• A linear equation in one variable is an equation of the form ax + b = c, where a, b, and c are constants, and x is the variable.
• To solve a linear equation, the goal is to isolate the variable x on one side of the equation.
• Steps to solve a linear equation:
  1. Add or subtract the same value to both sides of the equation to eliminate any constants.
  2. Multiply or divide both sides of the equation by a coefficient (a value multiplied by the variable) to eliminate any coefficients.
  3. Simplify the equation to get the final solution.

Linear Equation Word Problems

• Linear equation word problems involve using linear equations to model real-world scenarios.
• Steps to solve a linear equation word problem:
  1. Read and understand the problem statement.
  2. Identify the variable and the constants involved.
  3. Write a linear equation based on the information given.
  4. Solve the equation using the steps mentioned earlier.
  5. Interpret the solution in the context of the problem.

Examples of linear equation word problems:

• Cost problem: If it costs $x to buy x units of a product, and the total cost is $120 when buying 8 units, how much does one unit cost?
• Distance problem: If a car travels at a speed of x miles per hour, and it travels 240 miles in 4 hours, how fast was the car traveling?

Equation Manipulation

• Equation manipulation involves rearranging an equation to get it into a different form.
• Addition and subtraction property: The same value can be added or subtracted to both sides of an equation without changing its solution.
• Multiplication and division property: Both sides of an equation can be multiplied or divided by the same non-zero value without changing its solution.
• Equivalent equations: Two equations that have the same solution are called equivalent equations.

Examples of equation manipulation:

• Simplifying an equation: 2x + 3 = 7 can be simplified to 2x = 4 by subtracting 3 from both sides.
• Rearranging an equation: x - 2 = 5 can be rearranged to x = 7 by adding 2 to both sides.

Solving Linear Equations

• A linear equation in one variable is of the form ax + b = c, where a, b, and c are constants, and x is the variable.
• The goal is to isolate the variable x on one side of the equation.
• Steps to solve a linear equation:
  • Add or subtract the same value to both sides to eliminate constants.
  • Multiply or divide both sides by a coefficient to eliminate coefficients.
  • Simplify the equation to get the final solution.

Linear Equation Word Problems

• Linear equation word problems involve modeling real-world scenarios with linear equations.
• Steps to solve a linear equation word problem:
  • Read and understand the problem statement.
  • Identify the variable and constants involved.
  • Write a linear equation based on the given information.
  • Solve the equation using the steps mentioned earlier.
  • Interpret the solution in the context of the problem.

Equation Manipulation

• Equation manipulation involves rearranging an equation to get it into a different form.
• The addition and subtraction property allows adding or subtracting the same value to both sides of an equation.
• The multiplication and division property allows multiplying or dividing both sides of an equation by the same non-zero value.
• Equivalent equations have the same solution.
• Examples of equation manipulation:
  • Simplifying an equation: 2x + 3 = 7 can be simplified to 2x = 4.
  • Rearranging an equation: x - 2 = 5 can be rearranged to x = 7.

Solving Linear Equations

Definition

• A linear equation in one variable is an equation in the form ax + b = 0, where a and b are constants, and x is the variable.

Steps to Solve a Linear Equation

• Simplify the equation by combining like terms and removing parentheses.
• Isolate the variable by adding or subtracting the same value to both sides to get rid of the constant term, and then dividing both sides by the coefficient of the variable.
• Check the solution by plugging it back into the original equation to verify that it is true.

Examples

• Solve 2x + 3 = 7 by subtracting 3 from both sides, then dividing both sides by 2, resulting in x = 2.
• Solve x - 4 = 9 by adding 4 to both sides, resulting in x = 13.

Special Cases

• If the equation is a contradiction (e.g., 2 = 3), there is no solution.
• If the equation is an identity (e.g., 2x = 2x), the equation is true for all values of x.

Word Problems

• Read the problem and identify the unknown variable.
• Write an equation based on the information given.
• Solve the equation using the steps above.
• Check the solution to ensure it makes sense in the context of the problem.

Learn how to solve linear equations in one variable by isolating the variable x on one side of the equation. Steps include adding or subtracting constants and multiplying or dividing by coefficients.