Question

simplifying algebraic expressions

1. $5x + 9x$
2. $2x^2 + 3x - x^2 + 6x$
3. $4y - 7x - 12y + 3x$
4. $12 + 5(x - 1)$
5. $12 - 5(x - 1)$
6. $4(6 - x) + 3(x - 8)$

solving algebraic equations

1. $8x - 2 = -9 + 7x$
2. $-8 = -(x + 4)$
3. $-(7 - 4x) = 9$
4. $x - 1 = 5x + 3x - 8$
5. $2(4x - 3) - 8 = 4 + 2x$
6. $3x - 5 = -8(6 + 5x)$

Explanation:

Let's solve these algebraic problems one by one. We'll start with the simplifying algebraic expressions section (problems 20 - 25) and then move to solving algebraic equations (problems 26 - 31).

Simplifying Algebraic Expressions

Problem 20: \( 5x + 9x \)

Step 1: Combine like terms

Since both terms have the variable \( x \), we can add their coefficients. The coefficient of \( 5x \) is 5, and the coefficient of \( 9x \) is 9. So we add 5 and 9.
\( 5x + 9x = (5 + 9)x \)

Step 2: Calculate the sum of coefficients

\( 5 + 9 = 14 \), so \( (5 + 9)x = 14x \)

Step 1: Combine like terms for \( x^2 \) terms

The \( x^2 \) terms are \( 2x^2 \) and \( -x^2 \). Combine their coefficients: \( 2 - 1 = 1 \), so \( 2x^2 - x^2 = x^2 \)

Step 2: Combine like terms for \( x \) terms

The \( x \) terms are \( 3x \) and \( 6x \). Combine their coefficients: \( 3 + 6 = 9 \), so \( 3x + 6x = 9x \)

Step 3: Combine the results

Put the combined \( x^2 \) term and \( x \) term together: \( x^2 + 9x \)

Step 1: Combine like terms for \( y \) terms

The \( y \) terms are \( 4y \) and \( -12y \). Combine their coefficients: \( 4 - 12 = -8 \), so \( 4y - 12y = -8y \)

Step 2: Combine like terms for \( x \) terms

The \( x \) terms are \( -7x \) and \( 5x \). Combine their coefficients: \( -7 + 5 = -2 \), so \( -7x + 5x = -2x \)

Step 3: Combine the results

Put the combined \( y \) term and \( x \) term together: \( -8y - 2x \) (or \( -2x - 8y \))

Answer:

\( 14x \)

Problem 21: \( 2x^2 + 3x - x^2 + 6x \)