Solving Equations

Solving linear equations is a fundamental skill in algebra. Linear equations are equations where the highest power of the variable is 1. In this lesson, we'll cover techniques for solving linear equations, including one-step and two-step equations.

One-Step Equations:

A one-step equation is an equation that can be solved in a single operation. The goal is to isolate the variable on one side of the equation. Here are some common one-step equations:

• Addition and Subtraction Equations:

  • To solve equations of the form "x + a = b" or "x - a = b," isolate x by performing the inverse operation (subtracting or adding a) on both sides of the equation.

• Example 1: Solve for x in the equation "x + 4 = 10."

  • Subtract 4 from both sides: x + 4 - 4 = 10 - 4.

  • Simplify: x = 6.

• Example 2: Solve for y in the equation "y - 8 = 5."

  • Add 8 to both sides: y - 8 + 8 = 5 + 8.

  • Simplify: y = 13.

• Multiplication and Division Equations:

  • To solve equations of the form "ax = b" or "x/a = b," isolate x by performing the inverse operation (dividing by a or multiplying by the reciprocal of a) on both sides of the equation.

• Example 3: Solve for x in the equation "3x = 15."

  • Divide by 3 on both sides: (3x) / 3 = 15 / 3.

  • Simplify: x = 5.

• Example 4: Solve for z in the equation "z/2 = 6."

  • Multiply by 2 on both sides: (z/2) * 2 = 6 * 2.

  • Simplify: z = 12.

Two-Step Equations:

A two-step equation requires two operations to isolate the variable. Here's how to solve two-step equations:

• Isolate the Variable: First, use addition or subtraction to move terms to one side of the equation, isolating the variable.

Example 5: Solve for x in the equation "2x - 5 = 11."

• • Add 5 to both sides: 2x - 5 + 5 = 11 + 5.

  • Simplify: 2x = 16.

• Apply the Inverse Operation: Second, apply the inverse operation (division or multiplication) to isolate the variable completely.

  • Divide by 2 on both sides: (2x) / 2 = 16 / 2.

  • Simplify: x = 8.

Practice and Application:

• Practice solving various one-step and two-step equations.

• Solve real-world problems using equations to represent relationships.

• Challenge yourself with equations that involve fractions or decimals to strengthen your problem-solving skills.

Solving linear equations is a fundamental algebraic skill used in a wide range of applications. By mastering these techniques, you'll be better equipped to tackle more complex equations and solve real-world problems.