Question

topic 4: simplifying expressions
directions: simplify each expression.

1. $-6a + 15 - 14a + 8$
2. $7x - 2y + 6x - 18y$
3. $-y + 2y + 6 - 10y$
4. $9(x + 5)$
5. $-2(p - 10)$
6. $4(8a - 6)$
7. $-5(-3x + 8)$
8. $7(4a + 3b)$
9. $-3(4x + 5y)$
10. $3(4x + 1) - 9x - 12$
11. $-5(2 - 3m) + 8 - m$
12. $5 + 2(-6y - 2) - 5y$
13. $9 - 4(2p + 3) - 8p + 7$
14. $-11x - (x - 3) - 12 + 4x$
15. $-x + 2(3x + y) - 5y + 10x$

topic 5: factoring expressions
directions: factor each expression. if it cannot be factored, write \prime.\

1. $14x - 20$
2. $32y + 56$
3. $9x - 7$
4. $3x + 9y$
5. $8m + 35n$
6. $15x + 60y$

Explanation:

Step1: Identify the problem type

The problem involves simplifying algebraic expressions, which falls under the subfield of Algebra in Mathematics.

Step2: Analyze each expression (taking problem 13 as an example)

For the expression \(-6a + 15 - 14a + 8\), we combine like terms.

Step 2.1: Combine the \(a\) terms

\(-6a - 14a = (-6 - 14)a = -20a\)

Step 2.2: Combine the constant terms

\(15 + 8 = 23\)

Step 2.3: Write the simplified expression

So, \(-6a + 15 - 14a + 8 = -20a + 23\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(-20a + 23\), not \(-8a + 23\))

Step3: Analyze problem 14: \(7x - 2y + 6x - 18y\)

Step 3.1: Combine the \(x\) terms

\(7x + 6x = (7 + 6)x = 13x\)

Step 3.2: Combine the \(y\) terms

\(-2y - 18y = (-2 - 18)y = -20y\)

Step 3.3: Write the simplified expression

So, \(7x - 2y + 6x - 18y = 13x - 20y\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(13x - 20y\), not \(15x + 16y\))

Step4: Analyze problem 15: \(-y + 2y + 6 - 10y\)

Step 4.1: Combine the \(y\) terms

\(-y + 2y - 10y = (-1 + 2 - 10)y = -9y\)

Step 4.2: The constant term is \(6\)

Step 4.3: Write the simplified expression

So, \(-y + 2y + 6 - 10y = -9y + 6\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(-9y + 6\), not \(-11y + 6\))

Step5: Analyze problem 16: \(9(x + 5)\)

Using the distributive property \(a(b + c)=ab + ac\), we have:
\(9(x + 5)=9\times x + 9\times5 = 9x + 45\) (This one is correct as per the image)

Step6: Analyze problem 17: \(-2(p - 10)\)

Using the distributive property \(a(b - c)=ab - ac\), we have:
\(-2(p - 10)=-2\times p - (-2)\times10=-2p + 20\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(-2p + 20\), not \(-2p - 20\))

Step7: Analyze problem 18: \(4(8a - 6)\)

Using the distributive property \(a(b - c)=ab - ac\), we have:
\(4(8a - 6)=4\times8a - 4\times6 = 32a - 24\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(32a - 24\), not \(32a - 2\))

Step8: Analyze problem 19: \(-5(-3x + 8)\)

Using the distributive property \(a(-b + c)=-ab + ac\), we have:
\(-5(-3x + 8)=(-5)\times(-3x)+(-5)\times8 = 15x - 40\) (This one is correct as per the image)

Step9: Analyze problem 20: \(7(4a + 3b)\)

Using the distributive property \(a(b + c)=ab + ac\), we have:
\(7(4a + 3b)=7\times4a + 7\times3b = 28a + 21b\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(28a + 21b\), not \(28a + 216\))

Step10: Analyze problem 21: \(-3(4x + 5y)\)

Using the distributive property \(a(b + c)=ab + ac\), we have:
\(-3(4x + 5y)=-3\times4x + (-3)\times5y=-12x - 15y\) (This one is correct as per the image)

Step11: Analyze problem 22: \(3(4x + 1)-9x - 12\)

Step 11.1: Use the distributive property on \(3(4x + 1)\)

\(3(4x + 1)=3\times4x + 3\times1 = 12x + 3\)

Step 11.2: Combine like terms

\(12x + 3 - 9x - 12=(12x - 9x)+(3 - 12)=3x - 9\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(3x - 9\), not \(3x + 9\))

Step12: Analyze problem 23: \(-5(2 - 3m)+8 - m\)

Step 12.1: Use the distributive property on \(-5(2 - 3m)\)

\(-5(2 - 3m)=-5\times2 - (-5)\times3m=-10 + 15m\)

Step 12.2: Combine like terms

\(-10 + 15m + 8 - m=(15m - m)+(-10 + 8)=14m - 2\) (Note: The original answer in the image has a mistake here; the correct simplified form is \(14m - 2\), not \(-18 - 14m\))

Step13: Analyze problem 24: \(5 + 2(-6y - 2)-5y\)

Step 13.1: Use the distributive property on \(2(-6y - 2)\)

\(2(-6y - 2)=2\times(-6y)-2\times2=-12y - 4\)

Step 13.2: Combine like terms

\(5 - 12y - 4 - 5y=(-12y - 5y)+(5 - 4)=-17y + 1\)

Step14: Analyze problem 25: \(9 - 4(2p + 3)-8p + 7\)

Step 14.1: Use the distributive property on \(-4(2p + 3)\)

\(-4(2p + 3)=-4\times2p - 4\times3=-8p - 12\)

Step 14.2: Combine like terms

\(9 - 8p - 12 - 8p + 7=(-8p - 8p)+(9 - 12 + 7)=-16p + 4\)

Step15: Analyze problem 28: \(14x - 20\)

We find the greatest common factor (GCF) of \(14x\) and \(20\). The GCF of \(14\) and \(20\) is \(2\). So,
\(14x - 20 = 2(7x - 10)\) (This one is correct as per the image)

Step16: Analyze problem 29: \(32y + 56\)

The GCF of \(32\) and \(56\) is \(8\). So,
\(32y + 56 = 8(4y + 7)\) (This one is correct as per the image)

Step17: Analyze problem 30: \(9x - 7\)

Since there is no common factor other than \(1\) between \(9x\) and \(7\), the expression is prime (This one is correct as per the image)

Step18: Analyze problem 31: \(3x + 9y\)

The GCF of \(3x\) and \(9y\) is \(3\). So,
\(3x + 9y = 3(x + 3y)\) (Note: The original answer in the image has a mistake here; the correct factored form is \(3(x + 3y)\), not \(3(x + 3y)\) written as \(3(x + 3y)\) but the boxed part in the image seems to have a typo, maybe \(3(x + 3y)\) is correct)

Step19: Analyze problem 32: \(8m + 35n\)

Since there is no common factor other than \(1\) between \(8m\) and \(35n\), the expression is prime (This one is correct as per the image)

Step20: Analyze problem 33: \(15x + 60y\)

The GCF of \(15x\) and \(60y\) is \(15\). So,
\(15x + 60y = 15(x + 4y)\) (Note: The original answer in the image has a mistake here; the correct factored form is \(15(x + 4y)\), not \(15(x + 4y)\) written with a typo in the image)

Answer:

The correct simplified/factored forms for each problem (correcting the mistakes in the image) are:

1. \(-20a + 23\)

1. \(13x - 20y\)

1. \(-9y + 6\)

1. \(9x + 45\)

1. \(-2p + 20\)

1. \(32a - 24\)

1. \(15x - 40\)

1. \(28a + 21b\)

1. \(-12x - 15y\)

1. \(3x - 9\)

1. \(14m - 2\)

1. \(-17y + 1\)

1. \(-16p + 4\)

1. \(2(7x - 10)\)

1. \(8(4y + 7)\)

1. prime

1. \(3(x + 3y)\)

1. prime

1. \(15(x + 4y)\)