(4) Algebra: Linear equations 1 | Linear equations | Algebra I | Khan Academy

Introduction to Linear Equations

• Understanding basic linear equations and their solution methods.

Problem 1: 5x = 20

• The equation can be rephrased as: 5 times question mark = 20.

• This implies 5 multiplied by the variable x equals 20.

Systematic Solution Methods

• Commonly, one can deduce that 5 times 4 = 20 without much thought.

• However, to solve systematically:

  • Divide both sides by 5:

    • Left-hand side: 5x / 5 = x

    • Right-hand side: 20 / 5 = 4

    • Result: x = 4

  • Alternatively, multiply by the reciprocal (1/5):

    • This gives: x = 20 × 1/5, yielding x = 4.

Problem 2: -3/4 x = 10/13

• A more complex problem where we solve for x algebraically.

Steps to Solve

1. Identify the Coefficient:

   • The coefficient of x is -3/4.

2. Multiply by the Reciprocal:

   • The reciprocal of -3/4 is -4/3.

   • Multiply both sides by -4/3:

     • Left-hand side:

       • (-4/3) × (-3/4) = 1, leaving x.

     • Right-hand side:

       • (10/13) × (-4/3) = -40/39.

3. Result:

   • x = -40/39.

   • Alternate form: x = -1 and 1/39 (mixed number).

Verification of Solution

• Substitute back into the equation: (-3/4) × (-40/39) should equal 10/13.

  • Adjusting fractions gives:

    • -3 �d7 -40 = 120.

    • Followed by adjustments: 120 / 156 becomes 30/39 = 10/13 after simplification.

Problem 3: -5/6 x = 7/8

• Let's apply similar steps:

Steps to Solve

1. Identify the Coefficient:

   • The coefficient is -5/6.

2. Multiply by the Reciprocal:

   • The reciprocal of -5/6 is -6/5.

   • Multiply both sides by -6/5:

     • Left-hand side:

       • (-6/5) × (-5/6) = 1, resulting in x.

     • Right-hand side:

       • (7/8) × (-6/5) = -42/40.

3. Result:

   • Simplifying gives x = -21/20.

Verification of Solution

• Substitute back into the equation to ensure the correctness:

  • Calculation proceeds as: (-5/6) × (-21/20) results in 7/8 after correct adjustments.

Conclusion

• Effective methods to solve linear equations using division or multiplication by reciprocals.

• Encourage practice with various linear equations for mastery.