Example 1: Adding Algebraic Expressions Let's add $3x + 5y$ and$4x + 2y.$

Step 1: Identify like terms

• $3x$ and $4x$ are like terms because they both have the variable $x$.
• $5y$ and $2y$ are like terms because they both have the variable$y$.

Step 2: Combine the coefficients

• Combine $3x$ and $4x$: $3x + 4x = 7x$.
• Combine $5y$ and $2y$ : $5y + 2y = 7y$.

Step 3: Write the resulting expression

• The final expression is $7x + 7y$.

Example 2: Subtracting Algebraic Expressions Let's subtract $6a + 4b$ from $10a + 7b$.

Step 1: Identify like terms

• $10a$ and $6a$ are like terms because they both have the variable $a$.
• $7b$ and $4b$ are like terms because they both have the variable $b$.

Step 2: Combine the coefficients

• Subtract $6a$ from $10a$: $10a - 6a = 4a$.
• Subtract $4b$ from $7b$: $( 7b - 4b = 3b )$.

Step 3: Write the resulting expression

• The final expression is $4a + 3b .$

Exam Tip

• Double-check your signs: Remember to pay attention to whether you are adding or subtracting, as this will affect the sign of the final coefficients.
• Align like terms: When working with more complex expressions, it can be helpful to rewrite the terms so that like terms are aligned vertically, making it easier to add or subtract them correctly.

Practice Problems

1. Add $5x + 3y$ and $2x - 4y$.
2. Subtract $7a - 2b$ from $9a + 5b$.
3. Simplify $4m + 6n - 2m + 3n$.
4. Add $3x^2 + 2x - 1$ and $4x^2 - 3x + 5$.
5. Subtract $8p - 3q + 2r$ from $10p + 4q - 5r$.

Problem 1: Add $5x + 3y$ and $2x - 4y$.

Step 1: Identify like terms

• $5x$ and $2x$ are like terms because they both contain the variable $x$.

• $3y$ and $-4y$ are like terms because they both contain the variable $y$. Step 2: Combine the coefficients

• For the $x$ terms: $5x + 2x = 7x$.

• For the $y$ terms: $3y - 4y = -1y$ (which is also written as $-y$ . Step 3: Write the resulting expression

• The final expression is $7x - y .$ Explanation**:**

In this problem, we combined the coefficients of the like terms. $5x$ and $2x$ add together to make $7x$, and $3y$ minus $4y$results in $-y$.

Problem 2: Subtract $7a - 2b$ from $9a + 5b$.

Step 1: Identify like terms

• $9a$ and $7a$ are like terms because they both contain the variable $a$.

• $5b$and $-2b$ are like terms because they both contain the variable $b$. Step 2: Combine the coefficients

• For the $a$ terms: $9a - 7a = 2a$.

• For the $b$ terms: $5b - (-2b) = 5b + 2b = 7b$ (Remember that subtracting a negative is the same as adding). Step 3: Write the resulting expression

• The final expression is $( 2a + 7b )$. Explanation**:**

In this example, you need to subtract one expression from another. This is a bit more challenging because you have to remember to change the signs when subtracting. This type of question could appear in an exam to assess your understanding of subtraction with algebraic expressions.

Problem 3: Simplify $4m + 6n - 2m + 3n$.

Step 1: Identify like terms

• $4m$ and $-2m$are like terms because they both contain the variable $m$.

• $6n$ and $3n$ are like terms because they both contain the variable $n$. Step 2: Combine the coefficients

• For the $m$ terms: $4m - 2m = 2m$.

• For the $n$ terms: $6n + 3n = 9n$. Step 3: Write the resulting expression

• The final expression is $2m + 9n$. Explanation**:**

This problem involves simplifying an expression by combining like terms. It's important to carefully add and subtract the coefficients to ensure accuracy. Questions like these help you practice simplifying expressions, which is a key skill in algebra.

Problem 4: Add $3x^2 + 2x - 1$ and $4x^2 - 3x + 5$.

Step 1: Identify like terms

• $3x^2$ and $4x^2$are like terms because they both contain the variable $x^2$.

• $2x$ and $-3x$ are like terms because they both contain the variable $x$.

• $-1$ and $5$ are like terms because they are both constant terms (no variable). Step 2: Combine the coefficients

• For the$x^2$ terms: $3x^2 + 4x^2 = 7x^2$.

• For the $x$ terms: $2x - 3x = -1x$ (which is also written as $-x$).

• For the constant terms: $-1 + 5 = 4$. Step 3: Write the resulting expression

• The final expression is $( 7x^2 - x + 4 )$. Explanation**:**

This problem is slightly more advanced because it includes quadratic terms like $( x^2 )$.

Problem 5: Subtract $8p - 3q + 2r$ from $10p + 4q - 5r$.

Step 1: Identify like terms

• $10p$ and $8p$ are like terms because they both contain the variable $p$.

• $4q$ and $-3q$ are like terms because they both contain the variable $q$.

• $-5r$ and $2r$ are like terms because they both contain the variable $r$. Step 2: Combine the coefficients

• For the $p$ terms: $10p - 8p = 2p$.

• For the $q$ terms: $4q - (-3q) = 4q + 3q = 7q$ (Remember that subtracting a negative is the same as adding).

• For the $r$ terms: $-5r - 2r = -7r$. Step 3: Write the resulting expression

• The final expression is $( 2p + 7q - 7r )$. Explanation**:**

This problem involves subtracting one algebraic expression from another. It's important to carefully handle the signs, especially when subtracting.