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

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.