Exercise solution 4, page 482 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) Like terms can be combined, but unlike terms cannot. In algebra, you may add or subtract terms only when they have exactly the same variable part. For example, \(7x\) and \(-3x\) are like terms, so they can be combined. But \(7x\) and \(4y\) are not like terms, so they must stay separate.

\[ \begin{aligned} 7x - 3x + 4y &= 4x + 4y \end{aligned} \]

This is because the \(x\)-terms combine with each other, and the \(y\)-term stays as it is.

B) Coefficients are the numbers in front of the variables. When combining like terms, work with the coefficients and keep the variable part unchanged. For instance, in \(9a - 5a\), the coefficients are \(9\) and \(-5\).

\[ \begin{aligned} 9a - 5a &= 4a \end{aligned} \]

The variable is still \(a\), so only the numbers in front are added or subtracted.

C) Powers must also match for terms to be like terms. Terms such as \(x\) and \(x^2\) are not like terms, because their powers are different. So they cannot be combined into a single term.

For example, \(6m^2 + 3m\) cannot be simplified further by combining the two terms. They are different kinds of terms.

However, terms with the same variable and the same power can be combined:

\[ \begin{aligned} 8p^2 - 5p^2 &= 3p^2 \end{aligned} \]

D) Be careful not to change the power when simplifying. Adding or subtracting like terms does not change the exponent. For example, \(4x^2 + 2x^2\) does not become \(6x^4\). The correct simplification is:

\[ \begin{aligned} 4x^2 + 2x^2 &= 6x^2 \end{aligned} \]

You add the coefficients, but the variable part \(x^2\) stays the same.

E) Constant terms can be combined with other constants. A constant is a number with no variable. So numbers such as \(8\) and \(-11\) are like terms and can be combined.

\[ \begin{aligned} 8 - 11 &= -3 \end{aligned} \]

In a longer expression, combine variable terms with variable terms, and constant terms with constant terms.

F) Simplify systematically by grouping similar terms. A good method is to collect all the \(x\)-terms together, all the \(y\)-terms together, and all the constants together. This helps avoid mistakes.

For example:

\[ \begin{aligned} 5x + 3y - 2x - y + 9 - 4 &= (5x - 2x) + (3y - y) + (9 - 4) \\ &= 3x + 2y + 5 \end{aligned} \]

This step-by-step structure makes it easier to see which terms can be combined.

G) Check each term one by one. When looking for an error in a simplification, ask yourself:

Do these terms have the same variable?

Do they have the same power?

Is this a constant term?

If the answer is no, then the terms should not be combined. This is the key idea needed to spot mistakes in algebraic simplifications.