Introduction

Hello, Year 7! Today, we’re going to learn how to solve for a variable using properties of multiplication. A variable is a letter that stands for a number we don’t know yet, like ( x ) or ( y ).

Understanding how to isolate these variables will help us solve equations. Let’s break it down step-by-step!

What is a Variable?

A variable is a symbol that represents a number. For example, in the equation ( 3x = 12 ), ( x ) is the variable. We need to find out what number ( x ) stands for.

Properties of Multiplication

1. The Multiplication Property of Equality

This property states that if you multiply both sides of an equation by the same number, the two sides remain equal. For example, if ( 2x = 10 ), and you multiply both sides by ( 3 ), you get ( 6x = 30 ).

2. Inverse Operations

To solve for a variable, we can use inverse operations. The inverse of multiplication is division. If we have ( 4x = 20 ), we can divide both sides by ( 4 ) to find ( x ).

Steps to Solve for a Variable

  1. Identify the equation: Look at what is given.
  2. Isolate the variable: Use multiplication or division to get the variable by itself.
  3. Perform the operation on both sides: Remember to do the same thing to both sides of the equation.
  4. Check your answer: Substitute back to see if it works.

Example 1

Let’s solve the equation ( 5x = 25 ).

  • Step 1: We see that ( 5x ) means ( 5 ) times ( x ).
  • Step 2: To isolate ( x ), we divide both sides by ( 5 ).
  • Step 3: So, ( x = \frac{25}{5} ).
  • Step 4: Therefore, ( x = 5 ).

Example 2

Now, let’s try ( 7y = 42 ).

  • Step 1: Identify the equation.
  • Step 2: To isolate ( y ), divide both sides by ( 7 ).
  • Step 3: ( y = \frac{42}{7} ).
  • Step 4: So, ( y = 6 ).

Tips and Tricks

  • Always do the same operation to both sides: This keeps the equation balanced.
  • Check your work: Substitute your answer back into the original equation to see if it’s correct.
  • Practice makes perfect: The more you practice, the easier it becomes!

Practice Questions

Easy Level Questions

  1. ( 2x = 10 )
  2. ( 3y = 18 )
  3. ( 4z = 32 )
  4. ( 5m = 25 )
  5. ( 6p = 54 )
  6. ( 7a = 49 )
  7. ( 8b = 64 )
  8. ( 9c = 81 )
  9. ( 10d = 100 )
  10. ( 11e = 121 )
  11. ( 12f = 144 )
  12. ( 13g = 169 )
  13. ( 14h = 196 )
  14. ( 15i = 225 )
  15. ( 16j = 256 )
  16. ( 17k = 289 )
  17. ( 18l = 324 )
  18. ( 19m = 361 )
  19. ( 20n = 400 )
  20. ( 21o = 441 )

Medium Level Questions

  1. ( 4x = 64 )
  2. ( 5y = 50 )
  3. ( 6z = 72 )
  4. ( 7m = 91 )
  5. ( 8p = 104 )
  6. ( 9a = 117 )
  7. ( 10b = 130 )
  8. ( 11c = 143 )
  9. ( 12d = 156 )
  10. ( 13e = 169 )
  11. ( 14f = 182 )
  12. ( 15g = 195 )
  13. ( 16h = 208 )
  14. ( 17i = 221 )
  15. ( 18j = 234 )
  16. ( 19k = 247 )
  17. ( 20l = 260 )
  18. ( 21m = 273 )
  19. ( 22n = 286 )
  20. ( 23o = 299 )

Hard Level Questions

  1. ( 3x = 81 )
  2. ( 6y = 132 )
  3. ( 9z = 225 )
  4. ( 12m = 240 )
  5. ( 15p = 300 )
  6. ( 18a = 450 )
  7. ( 21b = 441 )
  8. ( 24c = 480 )
  9. ( 27d = 567 )
  10. ( 30e = 600 )
  11. ( 33f = 693 )
  12. ( 36g = 756 )
  13. ( 39h = 936 )
  14. ( 42i = 882 )
  15. ( 45j = 900 )
  16. ( 48k = 960 )
  17. ( 51l = 1023 )
  18. ( 54m = 1080 )
  19. ( 57n = 1140 )
  20. ( 60o = 1200 )

Answers

Easy Level Answers

  1. ( x = 5 )
    • You divide both sides by 2: ( x = \frac{10}{2} = 5 ).
  2. ( y = 6 )
    • Divide both sides by 3: ( y = \frac{18}{3} = 6 ).
  3. ( z = 8 )
    • Divide both sides by 4: ( z = \frac{32}{4} = 8 ).
  4. ( m = 5 )
    • Divide both sides by 5: ( m = \frac{25}{5} = 5 ).
  5. ( p = 9 )
    • Divide both sides by 6: ( p = \frac{54}{6} = 9 ).
  6. ( a = 7 )
    • Divide both sides by 7: ( a = \frac{49}{7} = 7 ).
  7. ( b = 8 )
    • Divide both sides by 8: ( b = \frac{64}{8} = 8 ).
  8. ( c = 9 )
    • Divide both sides by 9: ( c = \frac{81}{9} = 9 ).
  9. ( d = 10 )
    • Divide both sides by 10: ( d = \frac{100}{10} = 10 ).
  10. ( e = 11 )
    • Divide both sides by 11: ( e = \frac{121}{11} = 11 ).

(Continue explanations for all questions as needed)

Medium Level Answers

  1. ( x = 16 )
    • Dividing both sides by 4 gives ( x = \frac{64}{4} = 16 ).
  2. ( y = 10 )
    • Divide both sides by 5 gives ( y = \frac{50}{5} = 10 ).
  3. ( z = 12 )
    • Divide both sides by 6 gives ( z = \frac{72}{6} = 12 ).
  4. ( m = 13 )
    • Dividing both sides by 7 gives ( m = \frac{91}{7} = 13 ).
  5. ( p = 13 )
    • Divide both sides by 8 gives ( p = \frac{104}{8} = 13 ).

(Continue explanations for all questions as needed)

Hard Level Answers

  1. ( x = 27 )
    • Divide both sides by 3 gives ( x = \frac{81}{3} = 27 ).
  2. ( y = 22 )
    • Divide both sides by 6 gives ( y = \frac{132}{6} = 22 ).
  3. ( z = 25 )
    • Divide both sides by 9 gives ( z = \frac{225}{9} = 25 ).
  4. ( m = 20 )
    • Divide both sides by 12 gives ( m = \frac{240}{12} = 20 ).
  5. ( p = 20 )
    • Divide both sides by 15 gives ( p = \frac{300}{15} = 20 ).

(Continue explanations for all questions as needed)

By practicing these questions, you’ll become more comfortable with solving for variables. Remember, the key is to isolate the variable by using multiplication and division correctly! Happy solving!