Solving Equations Revision Notes for Edexcel GCSE Maths

Solving equations

What is solving equations?

Solving equations is one of the most fundamental skills in algebra. The basic idea is beautifully simple: you keep rearranging the equation until you end up with x = number. Think of it as unpacking a mathematical puzzle where you need to isolate the unknown variable on one side of the equation.

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There are two main approaches you can use when rearranging equations. The first is the "same to both sides" method, where whatever operation you perform on one side of the equation, you must also perform on the other side to keep it balanced. The second approach involves "doing the opposite when you cross the equals sign" - if you move a term from one side to the other, you change its sign or operation.

Basic method: Rearrange until you have x = number

The easiest equations to solve are those with a simple mixture of x-terms and numbers. Your goal is to rearrange the equation so that all the x-terms are on one side and all the numbers are on the other side.

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Your primary objective is to rearrange the equation so that all the x-terms are on one side and all the numbers are on the other side. This separation is the key to solving any linear equation successfully.

Here's the straightforward process:

First, rearrange the equation so that all the x-terms are on one side and all the numbers are on the other
Combine any like terms where you can
Then divide both sides by the number that's multiplying x to find the value of x

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Solve: $5x + 4 = 8x - 5$

Step 1: Add $5$ to both sides

$5x + 4 + 5 = 8x - 5 + 5$

$5x + 9 = 8x$

Step 2: Subtract $5x$ from both sides

$5x + 9 - 5x = 8x - 5x$

$9 = 3x$

Step 3: Divide both sides by $3$

$\dfrac{9}{3} = \dfrac{3x}{3}$

$3 = x$

Shortened working (once you're confident):

$5x + 9 = 8x$

$9 = 3x$

$3 = x$

Once you become comfortable with this method, you can use shortened working and don't need to write out every single step in full detail.

Dealing with brackets in equations

When your equation contains brackets, you need to multiply them out before you can start rearranging. This is a crucial first step that many students forget, but it makes the rest of the solving process much smoother.

The process is:

Multiply out any brackets first, before rearranging
Then solve the equation in the same way as shown above

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Always multiply out brackets first before attempting to rearrange the equation. This is a step that many students forget, but it's essential for solving the equation correctly.

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Worked Example: Solving with Brackets

If you have an equation like $3(3x - 2) = 5x + 10$ , you would first expand the bracket:

$3(3x - 2) = 5x + 10$

$9x - 6 = 5x + 10$

Then proceed with the normal solving method.

Dealing with fractions in equations

Fractions can make equations look much more complicated than they actually are, so it's best to get rid of them before you do anything else. The key principle is to multiply every term in the equation by whatever's on the bottom of the fraction.

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Fractions make everything more complicated, so you need to eliminate them before doing anything else. This should always be your first step when dealing with equations containing fractions.

Here's how to handle fractions:

Fractions make everything more complicated, so you need to eliminate them before doing anything else
To get rid of fractions, multiply every term of the equation by whatever's on the bottom of the fraction
If there are two fractions with different denominators, you'll need to multiply by both denominators

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Example 1

Solve: $\dfrac{x + 2}{4} = 4x - 7$

Step 1: Multiply every term by $4$ to remove the fraction

$4 \cdot \dfrac{x + 2}{4} = 4(4x) - 4(7)$

$x + 2 = 16x - 28$

Step 2: Subtract $x$ from both sides

$2 + 28 = 16x - x$

$30 = 15x$

Step 3: Divide both sides by $15$

$\dfrac{30}{15} = \dfrac{15x}{15}$

$2 = x$

Example 2

Solve: $\dfrac{3x + 5}{2} = \dfrac{4x + 10}{3}$

Step 1: Multiply everything by $2$ , then by $3$

$\dfrac{2 \times 3 \times (3x + 5)}{2} = \dfrac{2 \times 3 \times (4x + 10)}{3}$

Step 2: Simplify both sides

$3(3x + 5) = 2(4x + 10)$

Step 3: Expand brackets

$9x + 15 = 8x + 20$

Step 4: Solve for $x$

$9x - 8x = 20 - 15$

$x = 5$

The beauty of this method is that once you've eliminated the fractions, you're left with a much simpler equation that you can solve using the standard methods.

The comprehensive 6-step method

Once you understand the basics of solving equations, you can use this systematic 6-step method to tackle any equation, no matter how complex it might look. This method ensures you don't miss any important steps and helps you work through problems in a logical order.

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This comprehensive method works for virtually any linear equation you'll encounter. The key is to work through it systematically and not skip steps, even if some seem unnecessary for simpler equations.

Here's the complete method to follow (just ignore any steps that don't apply to your particular equation):

Get rid of any fractions - multiply every term by the denominators
Multiply out any brackets - expand all bracketed expressions
Collect all the x-terms on one side and all number terms on the other - rearrange to separate variables and constants
Reduce it to the form ' $Ax = B$ ' - combine like terms so you have a coefficient times x equals a number
Finally divide both sides by A - this gives you ' $x =$ ' and that's your answer
If you had ' $x^2 =$ ' instead, square root both sides - remember this gives you both positive and negative solutions

Dealing with squares and square roots

Sometimes you might encounter an equation where you end up with $x^2$ rather than just $x$ . When this happens, you need to take the square root of both sides to find the value of x.

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There's one very important thing to remember about square roots: whenever you take the square root of a number, the answer can be both positive and negative. This is because both positive and negative numbers give the same result when squared.

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Worked Example: Square Root Solutions

If you have $x^2 = 25$ , then $x = \pm 5$ (which means $x = +5$ or $x = -5$ ).

This is because $5^2 = 25$ and $(-5)^2 = 25$ . Your calculator will only show the positive answer, but mathematically, both solutions are correct.

Checking your answers

Here's a valuable final tip that can save you marks in exams: you can always check your answer by substituting it back into both sides of the original equation. If both sides give you the same number, then you know your solution is correct.

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This checking method is particularly useful for catching arithmetic errors or mistakes in your working. It's a good habit to develop and only takes a few seconds to verify your solution.

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Key Points to Remember:

The fundamental principle of solving equations is to keep rearranging until you have $x = \text{number}$
Always use the "same to both sides" approach - whatever you do to one side, you must do to the other
Deal with fractions first by multiplying every term by the denominator(s)
Multiply out brackets before attempting to rearrange the equation
When you get $x^2 = \text{number}$ , remember that $x = \pm\sqrt{\text{number}}$ (both positive and negative solutions)
Always check your answer by substituting it back into the original equation

Solving Equations (Edexcel GCSE Maths): Revision Notes