# 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**.