https://file.aiquickdraw.com/mathbot/file/1727272798858uc2mr9vh.jpg

From the image, the topic is "Manipulasi Aljabar" or "Algebraic Manipulation" in Indonesian. It explains algebraic manipulation as a process involving operations on algebraic forms that maintain symmetry or equality. These manipulations include expansions, multiplication, exponentiation, factorization, and simplifying algebraic forms.

The section shows three algebraic problems, and they ask for the value of \( a \) or \( t \) that satisfies the given equations.

Let's focus on solving these three examples:

### (1) Solve for \( a \):
\[
4(2a - 3) - 2(a - 4) = 3(a - 3) - 1
\]

Expand both sides:
\[
4(2a - 3) = 8a - 12
\]
\[
-2(a - 4) = -2a + 8
\]
So, the left-hand side becomes:
\[
8a - 12 - 2a + 8 = 6a - 4
\]

On the right-hand side, expand:
\[
3(a - 3) = 3a - 9
\]
\[
3a - 9 - 1 = 3a - 10
\]

Now equate both sides:
\[
6a - 4 = 3a - 10
\]

Simplify by subtracting \( 3a \) from both sides:
\[
3a - 4 = -10
\]

Add 4 to both sides:
\[
3a = -6
\]

Finally, divide by 3:
\[
a = -2
\]

### (2) Solve for \( a \):
\[
\frac{3}{a - 2} = \frac{4}{3a + 4}
\]

Cross-multiply:
\[
3(3a + 4) = 4(a - 2)
\]

Expand both sides:
\[
9a + 12 = 4a - 8
\]

Subtract \( 4a \) from both sides:
\[
5a + 12 = -8
\]

Subtract 12 from both sides:
\[
5a = -20
\]

Divide by 5:
\[
a = -4
\]

### (3) Solve for \( t \):
\[
\frac{\sqrt{t + 3}}{\sqrt{t}} = 2
\]

Multiply both sides by \( \sqrt{t} \):
\[
\sqrt{t + 3} = 2\sqrt{t}
\]

Square both sides:
\[
t + 3 = 4t
\]

Subtract \( t \) from both sides:
\[
3 = 3t
\]

Divide by 3:
\[
t = 1
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. What are some common methods for solving quadratic equations?
2. How do you simplify complex fractions in algebraic expressions?
3. What is the difference between factoring and expanding in algebra?
4. How can linear equations be used to model real-world problems?
5. Why is it important to maintain balance on both sides of an equation?

### Tip:
Always check your solutions by substituting the values back into the original equations to ensure they satisfy the equations.

Cari nilai a yang memenuhi:
(1) 4(2a - 3) - 2(a - 4) = 3(a - 3) - 1
(2) 3/(a - 2) = 4/(3a + 4)
(3) √(t + 3)/√t = 2

Explore the fundamentals of algebraic manipulation, focusing on expanding, factoring, and simplifying expressions to solve linear equations. This guide provides detailed solutions to example problems involving variables a and t, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation