Question

simplify the expression by combining like terms.

1. - 5x + 2x
2. 22m+( - 4m)
3. - 10x - 8x
4. 6x+12
5. - 32 + 18
6. 4 + m+3m
7. 9.2x - 7.9x
8. $\frac{7}{8}x+( - \frac{3}{4}x)$
9. 63n - 87n
10. $4x^{2}+8x^{2}-7$
11. $9a^{2}-5a^{2}-3$
12. 4j + 8 - 6j

simplify the expression.

1. 2(3x + 1)+x
2. - 2(3a - 5)+2a
3. ( - 4)(5m + 2)-3m
4. $2t^{2}+(3 + 5t)(4t)$
5. 2x(x + 4)-5x^{2}
6. - 2t(4t - 5)+( - 5t^{2})
7. 5 - 2(a + 8)
8. 5x - 2x(x + 7)
9. - y^{3}-6y(y^{2}-y)
10. geometry write an expression for the perimeter of the triangle shown below.
11. geometry write an expression for the perimeter of the trapezoid shown below.
12. geometry find the area of the shaded rectangle in two different ways. show how the results are related to the distributive property.
13. weight lifting a weight lifter puts an x - pound weight on each side of a bar. a weight 5 pounds heavier than the first is then added to both sides. finally, a weight 5 pounds heavier than the second weight is added to both sides. the expression 2x+(x + 5)+(x + 2(5)) models the total weight lifted. simplify the expression. what would the expression be if each added weight was 10 pounds heavier than the previous weight?

Explanation:

Step1: Combine like - terms for each expression

1. For \(-5x + 2x\), we have \((-5 + 2)x=-3x\).
2. For \(22m+( - 4m)\), \((22-4)m = 18m\).
3. For \(-10x-8x\), \((-10 - 8)x=-18x\).
4. For \(6x + 12\), it cannot be further combined as \(6x\) and \(12\) are not like - terms.
5. For \(-32+18\), \(-32 + 18=-14\).
6. For \(4 + m+3m\), \(4+(1 + 3)m=4 + 4m\).
7. For \(9.2x-7.9x\), \((9.2 - 7.9)x = 1.3x\).
8. For \(\frac{7}{8}x+( -\frac{3}{4}x)=\frac{7}{8}x-\frac{6}{8}x=\frac{7 - 6}{8}x=\frac{1}{8}x\).
9. For \(63n-87n\), \((63 - 87)n=-24n\).
10. For \(4x^{2}+8x^{2}-7\), \((4 + 8)x^{2}-7 = 12x^{2}-7\).
11. For \(9a^{2}-5a^{2}-3\), \((9 - 5)a^{2}-3 = 4a^{2}-3\).
12. For \(4j + 8-6j\), \((4-6)j + 8=-2j + 8\).
13. Expand \(2(3x + 1)+x\): First, \(2(3x + 1)=6x+2\), then \(6x + 2+x=(6 + 1)x+2=7x + 2\).
14. Expand \(-2(3a - 5)+2a\): First, \(-2(3a - 5)=-6a + 10\), then \(-6a+10 + 2a=(-6 + 2)a+10=-4a + 10\).
15. Expand \((-4)(5m + 2)-3m\): First, \((-4)(5m + 2)=-20m-8\), then \(-20m-8-3m=(-20 - 3)m-8=-23m-8\).
16. Expand \(2t^{2}+(3 + 5t)(4t)\): First, \((3 + 5t)(4t)=12t+20t^{2}\), then \(2t^{2}+12t + 20t^{2}=(2 + 20)t^{2}+12t=22t^{2}+12t\).
17. Expand \(2x(x + 4)-5x^{2}\): First, \(2x(x + 4)=2x^{2}+8x\), then \(2x^{2}+8x-5x^{2}=(2-5)x^{2}+8x=-3x^{2}+8x\).
18. Expand \(-2t(4t - 5)+(-5t^{2})\): First, \(-2t(4t - 5)=-8t^{2}+10t\), then \(-8t^{2}+10t-5t^{2}=(-8 - 5)t^{2}+10t=-13t^{2}+10t\).
19. Expand \(5-2(a + 8)\): First, \(-2(a + 8)=-2a-16\), then \(5-2a-16=-2a+(5 - 16)=-2a - 11\).
20. Expand \(5x-2x(x + 7)\): First, \(-2x(x + 7)=-2x^{2}-14x\), then \(5x-2x^{2}-14x=-2x^{2}+(5 - 14)x=-2x^{2}-9x\).
21. Expand \(-y^{3}-6y(y^{2}-y)\): First, \(-6y(y^{2}-y)=-6y^{3}+6y^{2}\), then \(-y^{3}-6y^{3}+6y^{2}=(-1 - 6)y^{3}+6y^{2}=-7y^{3}+6y^{2}\).
22. The perimeter of the triangle with sides \(2x\), \(2x\), and \(3x - 5\) is \(2x+2x+(3x - 5)=(2 + 2+3)x-5 = 7x-5\).
23. The perimeter of the trapezoid with sides \(4x\), \(3\), \(x + 3\), and \(9-2x\) is \(4x+3+(x + 3)+(9-2x)=(4 + 1-2)x+(3 + 3+9)=3x + 15\).
24. Method 1: The area of the large rectangle with length \(x\) and width \(5\) is \(A_1 = 5x\). The area of the non - shaded rectangle with length \(x - 3\) and width \(5\) is \(A_2=5(x - 3)=5x-15\). The shaded area \(A=5x-(5x - 15)=15\). Method 2: The shaded part can be considered as a rectangle with length \(3\) and width \(5\), so \(A = 3\times5=15\). Using the distributive property \(5x-5(x - 3)=5x-(5x-15)=5x-5x + 15 = 15\).
25. Simplify \(2[x+(x + 5)+(x + 10)]\): First, inside the brackets \(x+(x + 5)+(x + 10)=(1 + 1+1)x+(5 + 10)=3x + 15\). Then \(2(3x + 15)=6x+30\).

Answer:

1. \(-3x\)
2. \(18m\)
3. \(-18x\)
4. \(6x + 12\)
5. \(-14\)
6. \(4 + 4m\)
7. \(1.3x\)
8. \(\frac{1}{8}x\)
9. \(-24n\)
10. \(12x^{2}-7\)
11. \(4a^{2}-3\)
12. \(-2j + 8\)
13. \(7x + 2\)
14. \(-4a + 10\)
15. \(-23m-8\)
16. \(22t^{2}+12t\)
17. \(-3x^{2}+8x\)
18. \(-13t^{2}+10t\)
19. \(-2a - 11\)
20. \(-2x^{2}-9x\)
21. \(-7y^{3}+6y^{2}\)
22. \(7x-5\)
23. \(3x + 15\)
24. Area is \(15\) (using two methods and distributive property shown above)
25. Simplified expression is \(6x + 30\)