Solving Equations

Solving Equations

H1 Solving Linear Equations

  • Understand that the goal of solving an equation is to find the value of the variable that makes the equation true.
  • Apply the same operation to both sides of the equation to maintain the equality.
  • Subtract, multiply, or add the same value to both sides of the equation.
  • Example: For the equation 2x + 6 = 14, subtract 6 from both sides getting 2x = 8, then divide both sides by 2. The solution is x = 4

H1 Solving Equations Involving Brackets

  • Start by expanding the brackets. Ensure you understand the concept of distributive property.
  • Once the brackets are expanded, solve the equation as you would a normal linear equation.
  • Example: For the equation 2(x + 3) = 12, expand the bracket to get 2x + 6 = 12. Then solve for x by subtracting 6 and dividing by 2 to get x = 3.

H1 Solving Equations with Fractions

  • Multiply through by the denominator of the fraction to clear fractions
  • Then, solve the equation as you would a normal equation.
  • Example: For the equation 1/2x = 5, multiply through by 2 to get x = 10.

H1 Solving Quadratic Equations

  • To solve a quadratic equation, you can factor, complete the square or use the quadratic formula
  • Example with factoring: For x^2 - 5x + 6 = 0, the quadratic factors to (x- 3)(x - 2) = 0, and setting each bracket to zero gives the roots x = 3 and x = 2.
  • Example with quadratic formula: For the equation x^2 - 5x + 6 = 0, using the quadratic formula — which is x = [-b ± sqrt(b^2 - 4ac)] / 2a — taking a=1, b=-5, c=6. Solving this gives x = 3 and x = 2.

H1 Solving Equations with Unknowns on Both Sides

  • Your aim is to get the variable on one side only. To do this, perform operations that allow you to simplify the equation.
  • Example: For the equation 3x + 5 = 2x + 7, subtract 2x from both sides of the equation to get x + 5 = 7, then subtract 5 from both sides to get x = 2.

H1 Solving Simultaneous Equations

  • Simultaneous equations are solved by finding values of the variables that satisfy all of the given equations at the same time.
  • You can solve simultaneous equations by substitution or elimination method.
  • Example with substitution: Given the system of equations y = 2x and y = x + 3, substitute y from the first equation into second to get 2x = x + 3. Solving for x gives x = 3 and hence y = 6.

H1 Checking Solutions of an Equation

  • After finding a solution to an equation, it’s essential to check your answer by substituting it back into the original equation.
  • If the equation balances with the value you found for the variable, then you know your solution is correct.