Introduction to Two-Variable Equations

Today, we are going to learn about two-variable equations. These equations have two letters, usually ( x ) and ( y ). Each letter represents a number. Our goal is to find the value of ( x ) or ( y ) using what we know about the equation.

What is a Two-Variable Equation?

A two-variable equation looks like this:

y = 2x + 3

In this equation:

  • ( y ) is the output.
  • ( x ) is the input.
  • The equation shows how ( y ) changes when we change ( x ).

Key Rules

  1. Substitution: You can plug in a value for ( x ) or ( y ) to find the other variable.For example, if ( x = 2 ):y = 2(2) + 3 = 4 + 3 = 7So, when ( x = 2 ), ( y = 7 ).
  2. Finding Values: You can also rearrange the equation to solve for one variable in terms of the other.For example, if we want to express ( x ) in terms of ( y ):y = 2x + 3 \implies y – 3 = 2x \implies x = \frac{y – 3}{2}

Tips and Tricks

  • Graphing: Sometimes, it helps to graph the equation. Each pair of ( (x, y) ) values forms a point on the graph.
  • Checking Your Work: Always plug your found value back into the original equation to check if it works.
  • Practice: The more you practice, the easier it gets!

Example Problems

Let’s try a couple of examples together.

  1. Example 1: Find ( y ) when ( x = 4 ) in the equation ( y = 3x – 5 ).Substituting ( x ):y = 3(4) – 5 = 12 – 5 = 7So, ( y = 7 ).
  2. Example 2: Find ( x ) when ( y = 11 ) in the equation ( y = 4x + 1 ).Rearranging:11 = 4x + 1 \implies 10 = 4x \implies x = \frac{10}{4} = 2.5So, ( x = 2.5 ).

Practice Questions

Easy Level Questions

  1. Find ( y ) when ( x = 1 ): ( y = 2x + 2 )
  2. Find ( y ) when ( x = 3 ): ( y = x + 5 )
  3. Find ( y ) when ( x = 0 ): ( y = 4x – 1 )
  4. Find ( y ) when ( x = 2 ): ( y = 5x + 1 )
  5. Find ( y ) when ( x = 5 ): ( y = 6 – x )
  6. Find ( y ) when ( x = -1 ): ( y = 3x + 7 )
  7. Find ( y ) when ( x = 4 ): ( y = 10 – 2x )
  8. Find ( y ) when ( x = 6 ): ( y = x + 4 )
  9. Find ( y ) when ( x = 7 ): ( y = 3x – 10 )
  10. Find ( y ) when ( x = 2 ): ( y = 2x^2 + 1 )

Medium Level Questions

  1. Find ( y ) when ( x = 0 ): ( y = 2x^2 + 3x + 1 )
  2. Find ( y ) when ( x = 3 ): ( y = x^2 – 2x + 4 )
  3. Find ( y ) when ( y = 10 ): ( y = 5x + 5 )
  4. Find ( x ) when ( y = 8 ): ( y = 4x + 4 )
  5. Find ( x ) when ( y = 12 ): ( y = 3x + 6 )
  6. Find ( x ) when ( y = 0 ): ( y = 2x – 4 )
  7. Find ( y ) when ( x = 5 ): ( y = 2x^2 + 3x – 1 )
  8. Find ( x ) when ( y = 1 ): ( y = x^2 – 1 )
  9. Find ( y ) when ( x = 2 ): ( y = x^3 – 2x )
  10. Find ( x ) when ( y = 9 ): ( y = x^2 + 5x + 6 )

Hard Level Questions

  1. Find ( x ) when ( y = 7 ): ( y = 2x^2 – 3x + 4 )
  2. Find ( y ) when ( x = 3 ): ( y = x^2 – 5x + 6 )
  3. Find ( y ) when ( x = 1 ): ( y = 3x^3 + 2x^2 – x )
  4. Find ( y ) when ( x = -2 ): ( y = x^2 + 4x + 4 )
  5. Find ( x ) when ( y = 16 ): ( y = 4x^2 – 8x + 4 )
  6. Find ( y ) when ( x = 4 ): ( y = x^3 – x^2 + 2x – 1 )
  7. Find ( x ) when ( y = -3 ): ( y = -x^2 + 4x – 1 )
  8. Find ( y ) when ( x = -1 ): ( y = 2x^3 + 3x^2 – 4 )
  9. Find ( x ) when ( y = 0 ): ( y = 2x^2 + 3x + 1 )
  10. Find ( x ) when ( y = 5 ): ( y = x^2 + x – 6 )

Answers

Answers to Easy Level Questions

  1. ( y = 4 )
  2. ( y = 8 )
  3. ( y = -1 )
  4. ( y = 11 )
  5. ( y = 1 )
  6. ( y = 4 )
  7. ( y = 2 )
  8. ( y = 10 )
  9. ( y = 11 )
  10. ( y = 9 )

Answers to Medium Level Questions

  1. ( y = 1 )
  2. ( y = 7 )
  3. ( x = 1 )
  4. ( x = 1 )
  5. ( x = 2 )
  6. ( x = 2 )
  7. ( y = 66 )
  8. ( x = 0 )
  9. ( y = 2 )
  10. ( x = -3 )

Answers to Hard Level Questions

  1. ( x = 2 ) or ( x = 1 )
  2. ( y = 0 )
  3. ( y = 0 )
  4. ( y = 0 )
  5. ( x = 1 ) or ( x = 2 )
  6. ( y = 33 )
  7. ( x = 6 )
  8. ( y = -1 )
  9. ( x = \text{No Real Solutions} )
  10. ( x = 2 ) or ( x = -3 )

That’s all for today! Keep practising, and soon you’ll be finding values using two-variable equations with ease!