## Advanced ACT Math: Tricky Algebra Questions

The ACT Math section often includes challenging algebra questions that require a deep understanding of algebraic concepts and creative problem-solving skills. To excel in this section, it's crucial to be familiar with various algebraic techniques and strategies for tackling complex problems. Here are some advanced algebra topics commonly tested on the ACT Math section, along with examples of tricky algebra questions:

## 1. Systems of Equations:

#### Question 1:

```

If 2x + y = 10 and 3x - 2y = 4, what is the value of x + y?

```

#### Solution:

To solve this system of equations, we can use substitution or elimination. Let's use elimination:

```

2x + y = 10

3x - 2y = 4

```

Multiply the first equation by 2:

```

4x + 2y = 20

3x - 2y = 4

```

```

4x + 2y + 3x - 2y = 20 + 4

7x = 24

```

Substitute x back into the first equation to find y:

```

48/7 + y = 10

y = 70/7 - 48/7

y = 22/7

```

Finally, find the value of x + y:

```

x + y = 24/7 + 22/7

x + y = 46/7

```

#### Question 2:

```

If x^2 - 5x + 6 = 0, what are the solutions for x?

```

#### Solution:

To solve this quadratic equation, we can factor or use the quadratic formula. Let's factor:

```

x^2 - 5x + 6 = 0

(x - 2)(x - 3) = 0

```

Setting each factor equal to zero gives us the solutions:

```

x - 2 = 0  -->  x = 2

x - 3 = 0  -->  x = 3

```

## 3. Exponential and Logarithmic Equations:

#### Question 3:

```

If log(x) = 2, what is the value of x^2?

```

#### Solution:

To solve this problem, we need to understand the relationship between logarithmic and exponential functions:

```

log(x) = 2  -->  x = 10^2  -->  x = 100

```

Now, find the value of x^2:

```

x^2 = 100^2 = 10000

```

## 4. Functions and Function Notation:

#### Question 4:

```

If f(x) = x^2 + 3x - 2 and g(x) = 2x + 5, what is the value of f(g(2))?

```

#### Solution:

To find f(g(2)), we first need to find g(2) and then plug that result into f(x):

```

g(2) = 2(2) + 5 = 9

```

Now, plug g(2) into f(x):

```

f(g(2)) = f(9) = 9^2 + 3(9) - 2 = 81 + 27 - 2 = 106

```

## 5. Complex Numbers:

#### Question 5:

```

What is the product of (3 + 2i) and (4 - i)?

```

#### Solution:

To find the product of these complex numbers, use the distributive property:

```

(3 + 2i)(4 - i) = 3(4) + 3(-i) + 2i(4) + 2i(-i)

= 12 - 3i + 8i - 2i^2

= 12 + 5i - 2(-1)

= 12 + 5i + 2

= 14 + 5i

```

## Tips for Tackling Tricky Algebra Questions:

1. Understand the Concepts: Make sure you have a solid understanding of algebraic concepts and techniques before attempting advanced problems.

2. Practice Regularly: Practice solving a variety of algebraic problems to become familiar with different problem-solving strategies.

3. Stay Organized: Keep your work neat and organized, especially when dealing with multiple steps or complex equations.

4. Review Mistakes: Review any mistakes you make carefully and understand why you made them. Learning from your mistakes will help you avoid similar errors in the future.

5. Use Your Calculator Wisely: While calculators are allowed on the ACT Math section, they won't always be necessary for solving algebra problems. Use your calculator strategically and rely on mental math when possible.

## Conclusion

By mastering these tricky algebra concepts and practicing with similar questions, you'll be better equipped to tackle challenging algebra problems on the ACT Math section.