Solving Equations with Brackets

When you encounter equations in maths, you’ll often see brackets involved. Brackets serve a particular role in equations: they help clarify the order of operations.

Inside the brackets, operations are performed first. You’ll often come across the term ‘distributing’. In this context. distributing means you multiply the number outside the bracket with each term inside the bracket.

Removing the Brackets

Single Bracket

Let’s consider an example: 2(x + 3) = 10.

To remove the brackets, you’ll need to distribute the 2 across both x and 3. This turns the equation into 2x + 2 \times 3 = 10, simplifying to 2x + 6 = 10.

Multiple Brackets

For equations with more than one set of brackets, like 2(x - 3) + 3(x + 1) = 12, you distribute each number outside the brackets across the terms inside it.

Doing so results in 2x - 6 + 3x + 3 = 12. After distributing, combine like terms.

Combining Like Terms

In the example above, after distributing the numbers outside the brackets, you get 2x - 6 + 3x + 3 = 12. The next step is to combine like terms on each side of the equation, turning it into 5x = 15.

Solving the Equation

After you’ve removed brackets and combined like terms, it’s business as usual. Solve for the variable, x, just like you would for simpler equations.

In our example, 5x = 15, so x = 3. It’s always good to substitute your answer back into the original equation to make sure it’s correct.

Negative Numbers and Fractions Outside the Brackets

What happens if you have a negative number or a fraction outside the bracket? Let’s say you have -2(x + 3) = -8. The principle remains the same; distribute the -2 to each term inside the bracket to get -2x - 6 = -8. From there, solve for x as you normally would.

Practice Questions

Question 1:

Solve x + 3 = 7

Subtract 3 from both sides, and you get x = 4x = 4


Question 2:

Solve 3(x - 1) = 6

Distribute 3 to get 3x - 3 = 63x - 3 = 6

Add 3 to both sides, then divide by 3, and x = 3x = 3


Question 3:

Solve -2(x + 5) = -12

Distribute -2 to get -2x - 10 = -12-2x - 10 = -12

Add 10 to both sides, then divide by -2, and x = 1x = 1


Question 4:

Solve 2(x + 4) - 3 = 4(x - 1) + 1

Distribute to get 2x + 8 - 3 = 4x - 4 +12x + 8 - 3 = 4x - 4 +1

Combine like terms to get 2x + 5 = 4x - 32x + 5 = 4x - 3

Subtract 2x and add 3 to both sides to get 2x=82x=8

Therefore, x = 4x = 4

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